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Page 1: Pavement Evaluation Using Ground Penetrating … Evaluation Using Ground Penetrating Radar Technical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. MN/RC 2008-10

Take the steps...

Transportation Research

Research...Knowledge...Innovative Solutions!

2008-10

Pavement Evaluation UsingGround Penetrating Radar

Page 2: Pavement Evaluation Using Ground Penetrating … Evaluation Using Ground Penetrating Radar Technical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. MN/RC 2008-10

Technical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. MN/RC 2008-10 4. Title and Subtitle 5. Report Date

March 2008 6. Pavement Evaluation using Ground Penetrating Radar

7. Author(s) 8. Performing Organization Report No. Yuejian Cao, Bojan B. Guzina , Joseph F. Labuz 9. Performing Organization Name and Address 10. Project/Task/Work Unit No.

11. Contract (C) or Grant (G) No.

Department of Civil Engineering University of Minnesota 500 Pillsbury Dr. SE Minneapolis, Minnesota 55455

(c) 81655 (wo) 131

12. Sponsoring Organization Name and Address 13. Type of Report and Period Covered Final Report 14. Sponsoring Agency Code

Minnesota Department of Transportation 395 John Ireland Boulevard Mail Stop 330 St. Paul, Minnesota 55155 15. Supplementary Notes http://www.lrrb.org/PDF/200810.pdf 16. Abstract (Limit: 200 words) The objective of this project was to develop an efficient and accurate algorithm for the back analysis of pavement conditions measured by ground penetrating radar (GPR). In particular, more reliable information about the thickness of the asphalt concrete (AC) layer and the dielectric constants of the AC and base layers were obtained from the electromagnetic field measurements performed on roads using GPR. A brief introduction to the existing methodology for interpreting GPR images is reviewed, and the theory associated with electromagnetic wave propagation in layered structures is described. Utilizing the full waveform solution, algorithms for back analysis of pavement conditions were developed based on the artificial neural network approach and the frequency response function concept. Software called ''GopherGPR'' uses the GPR signal from one antenna to interpret the characteristics of the AC layer with no assumptions on material properties. Thus, the new technique has the capability of providing information not previously available.

17. Document Analysis/Descriptors 18. Availability Statement GPR thickness dielectric constant evaluation

No restrictions. Document available from: National Technical Information Services, Springfield, Virginia 22161

19. Security Class (this report) 20. Security Class (this page) 21. No. of Pages 22. Price Unclassified Unclassified 102

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PAVEMENT EVALUATION USING

GROUND PENETRATING RADAR

Final Report

Prepared by

Yuejian Cao, Bojan Guzina, and Joseph Labuz

Department of Civil Engineering

University of Minnesota

March 2008

Prepared forMinnesota Department of TransportationOffice of Materials and Road Research

1400 Gervais AvenueMaplewood, MN 55109

This report represents the results of research conducted by the authors and does not neces-sarily represent the views or policies of the Minnesota Department of Transportation and/orthe Center for Transportation Studies. This report does not contain a standard or specifiedtechnique.

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Acknowledgement

Partial support was provided by the Minnesota Department of Transportation. Specialthanks are extended to the Technical Advisory Panel, especially Dr. Shongtao Dai and Mr.Matthew Lebens, for sharing their expertise and knowledge of ground penetrating radar.Geophysical Survey Systems supplied some necessary information on data acquisition. Thisreport represents the results of research conducted by the authors and does not necessarilyrepresent the views or policies of the Minnesota Department of Transportation and/or theCenter for Transportation Studies.

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Contents

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 Background Information . . . . . . . . . . . . . . . . . . . 3

2.1 GPR applications . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 GPR waveform analysis . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 3 Correlation with Dielectric . . . . . . . . . . . . . . . . . . 10

3.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 113.1.1 Data from fine/coarse-grained soil . . . . . . . . . . . . . . . . . . . . 123.1.2 Data from MnDOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.3 Data from laboratory asphalt concrete sample . . . . . . . . . . . . . 16

3.2 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Data from fine and coarse-grained soil . . . . . . . . . . . . . . . . . 173.2.2 Field data from MnDOT . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.3 Data from laboratory asphalt concrete samples . . . . . . . . . . . . . 22

Chapter 4 Electromagnetic Waveform Analysis in Layered System . . . . 23

Chapter 5 Backcalculation Algorithm . . . . . . . . . . . . . . . . . . 26

5.1 Artificial neural network (ANN) . . . . . . . . . . . . . . . . . . . . 26

5.2 Frequency response function (FRF) . . . . . . . . . . . . . . . . . . 31

5.3 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.4 Single vs multiple sublayers in asphalt . . . . . . . . . . . . . . . . . 43

5.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Chapter 6 Implementation . . . . . . . . . . . . . . . . . . . . . . . 51

6.1 FRF patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.3 Implementation procedures . . . . . . . . . . . . . . . . . . . . . . 55

6.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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Chapter 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 60

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Appendix A: Data Set

Appendix B: Electromagnetic Analysis in Layered System

Appendix C: Manual for GPR interpretation

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List of Tables

Table 3.1 Fine-grained soil description . . . . . . . . . . . . . . . . . . . . 13Table 3.2 Coarse-grained soil description . . . . . . . . . . . . . . . . . . . 13Table 3.3 Dielectric Constants . . . . . . . . . . . . . . . . . . . . . . . . 14

Table 5.1 GPR antenna box inside spacings . . . . . . . . . . . . . . . . . . 36Table 5.2 Pavement properties with variable parameters. . . . . . . . . . . . . 36Table 5.3 H vs R-values . . . . . . . . . . . . . . . . . . . . . . . . . . 40Table 5.4 R-values vs H . . . . . . . . . . . . . . . . . . . . . . . . . . 43Table 5.5 Test runs of 3-layered system: 2 sublayers + base layer . . . . . . . . . 49

Table A-1 Data from the MnDOT . . . . . . . . . . . . . . . . . . . . . . A-1Table A-2 Data from lab asphalt sample . . . . . . . . . . . . . . . . . . . A-4Table A-3 Data from fine-grained soil : prismatic sample . . . . . . . . . . . . A-5Table A-4 Data from fine-grained soil : cylinder sample . . . . . . . . . . . . . A-5Table A-5 Data from coarse-grained soil . . . . . . . . . . . . . . . . . . . A-6

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List of Figures

Figure 1.1 GPR waveform in time history . . . . . . . . . . . . . . . . . . . 1

Figure 2.1 Asymptotic term Ga, residual term Gr and total kernel G . . . . . . . 8

Figure 3.1 Percometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 3.2 Voids and moisture absorption of mixtures . . . . . . . . . . . . . . 15Figure 3.3 Fine-grained soil: dielectric constant vs VWC . . . . . . . . . . . . 17Figure 3.4 Fine-grained soil: square root dielectric constant vs VWC . . . . . . . 18Figure 3.5 Coarse-grained: dielectric constant vs VWC . . . . . . . . . . . . . 19Figure 3.6 Coarse-grained: square root dielectric constant vs VWC . . . . . . . . 19Figure 3.7 Field asphalt concrete: square root dielectric constant vs VWC . . . . . 20Figure 3.8 Field asphalt concrete: dielectric constant vs VWC . . . . . . . . . . 20Figure 3.9 Field asphalt concrete: square root dielectric constant vs air voids. . . . 21Figure 3.10 Field asphalt concrete: dielectric constant vs air voids . . . . . . . . 21

Figure 4.1 Schematic of layered structure . . . . . . . . . . . . . . . . . . . 24

Figure 5.1 Node–computing unit . . . . . . . . . . . . . . . . . . . . . . . 27Figure 5.2 (a) Linear activation function, (b) Tan-sigmoid activation function . . . 28Figure 5.3 Three-layer feed-forward neural network . . . . . . . . . . . . . . . 29Figure 5.4 Principle of training neural network. . . . . . . . . . . . . . . . . 30Figure 5.5 Generality of function approximation on a noisy sine function . . . . . 31Figure 5.6 Areas under FRF . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 5.7 Spacings of GPR antenna box . . . . . . . . . . . . . . . . . . . 35Figure 5.8 GPR implementation configuration . . . . . . . . . . . . . . . . . 37Figure 5.9 Network with three outputs . . . . . . . . . . . . . . . . . . . . 39Figure 5.10 Linear regressions for (a) εA, (b) hA and (c) εB (H 31) . . . . . . . 41Figure 5.11 Network combined with three single output networks . . . . . . . . . 42Figure 5.12 Linear regressions for (a) εA, (b) hA and (c) εB (H 27) . . . . . . . 44Figure 5.13 Whole layer vs different layers in asphalt . . . . . . . . . . . . . . 45Figure 5.14 Two sublayers in asphalt . . . . . . . . . . . . . . . . . . . . . 47

Figure 6.1 The overlap of the signal from the feed point . . . . . . . . . . . . . 52Figure 6.2 Comparison of reflection between simulation and GPR survey . . . . . 54Figure 6.3 GPR survey of metal plate . . . . . . . . . . . . . . . . . . . . 54

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Figure 6.4 Two steps to setup project . . . . . . . . . . . . . . . . . . . . 56Figure 6.5 GopherGPR panel . . . . . . . . . . . . . . . . . . . . . . . . 58

Figure B-1 Schematic of layered structure . . . . . . . . . . . . . . . . . . . B-5Figure B-2 Reflection and transmission coefficients . . . . . . . . . . . . . . . B-9Figure C-1 GopherGPR panel . . . . . . . . . . . . . . . . . . . . . . . . C-2

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Executive Summary

The objective of this project was to develop an efficient and accurate algorithm for the backanalysis of pavement conditions measured by ground penetrating radar (GPR). In particular,more reliable information about the thickness of the asphalt concrete (AC) layer and thedielectric constants of the AC and base layers were obtained from the electromagnetic fieldmeasurements performed on roads using GPR.

A brief introduction to the existing methodology for interpreting GPR images is reviewed,and the theory associated with electromagnetic wave propagation in layered structures isdescribed. Utilizing the full waveform solution, algorithms for back analysis of pavementconditions were developed based on the artificial neural network approach and the frequencyresponse function concept. Software called ”GopherGPR” uses the GPR signal from oneantenna to interpret the characteristics of the AC layer with no assumptions on materialproperties. Thus, the new technique has the capability of providing information not previ-ously available.

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where the electromagnetic wave velocities vA (in the asphalt layer) and vB (in the base layer)are related to the dielectric constants εA and εB in each layer by,

vA c?εA

, vB c?εB

(1.2)

where c as the speed of light in vacuum.In the travel-time technique, the dielectric constant (ε) and travel time (∆t) are the only

information pertinent in determining the layer thickness, which is easy to implement. Thereare, however, quite a few instances of having difficulties to obtain these parameters. Forexample, it is possible that the in-situ dielectric constant values are unknown when doingthe survey. If the value of dielectric constant is taken from empirical knowledge, it willmost likely degrade the accuracy of the estimated layer thickness. In addition, each peak inthe temporal GPR record represents a pulse reflected back from the layer interface. Duringthe field survey, the peak may be overwhelmed by the ambient noise, thus multiplying thedifficulty of identifying the travel time between interfaces. As a consequence of the abovetwo sources of error, layer thickness computed in equation (1.1) cannot represent the truelayer structure. In estimation of the layer thickness, the state of art of GPR travel-timetechnique has an error around 7.5% compared to coring.

The objective of this project was to demonstrate the capabilities and limitations of groundpenetrating radar (GPR) for use in local road applications. The effectiveness of a GPR surveyis a function of site conditions, the equipment used, and experience of personnel interpretingthe results. In addition, not all site conditions are appropriate for GPR applications. GPR isa nondestructive field test that can provide a continuous profile of existing road conditions.GPR utilizes high-speed data collection at speeds up to 50 mph, thus requiring less trafficcontrol and resulting in greater safety. GPR has the potential to be used for a varietyof pavement applications, including measuring the thickness of asphalt pavement, base andsub-grade; assisting in the analysis of rutting mechanisms; calculating and verifying materialproperties; locating subsurface objects; detecting stripping and/or layer separation; detectingsubsurface moisture; and determining depth to near-surface bedrock and peat deposits.

In order to improve the performance of GPR survey, a new technique stems from thefull electromagnetic waveform analysis layered system has been developed. The detail ofthe EM model is reported in Chapter 4. The success of this layered EM wave model liesin its ability to reproduce the entire GPR scan (i.e., the waveform) including informationabout time and amplitude. These waveforms, which contain the information about the layerthickness and dielectric constant, are individually different. In other words, the waveformis labeled by its pattern called the frequency response function (FRF). With the aid of anartificial neural network (ANN) algorithm, it is further allowed to identify the connectionsbetween the pattern and the parameters of the waveform, as will be illustrated in Chapter5. The neural network has to be trained to establish a meaningful representation of theprofile and the GPR data. Once the neural network has been trained, the correspondinglayer properties can be obtained by inputting the FRF of the GPR response to the network.The implementation procedure is reported in Chapter 6.

2

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Chapter 2

Background Information

The examples of GPR application include the determination of layer thickness, voids detec-tion and moisture content measurement. The main adventures toward the interpretation ofGPR measurements, as is usually done in the context of GPR surveys, is the conventionaltravel-time analysis. This method is easy from the practical standpoint and has been widelyused but the limitation is also obvious, as will be discussed more in this chapter. To compare,the full electromagnetic waveform analysis, as another promising approach, will be brieflyintroduced as well.

2.1 GPR applications

Layer thickness

Since the introduction of the GPR to pavement engineering, its most successful applicationis to determine the pavement layer thickness. By “picking” the first peak of the reflectedelectromagnetic wave, the thickness is determined by the layer dielectric constant ε andtravel time t in nanoseconds.

hrin.s 11.8 t?ε

(2.1)

It is worth noting that the recorded travel time should be divided by two to represent a roundtrip, up and down. The dielectric constant, however, depends on the in-situ conditions.Obtaining cores for calibration could lead to a reduced error in estimation of layer thickness.However, it is apparently only applicable to a few spots due to the variation of the dielectricalong the road.

In practice, one may use the amplitudes of the reflected pulses to estimate the in-situdielectric constant. Given that the dielectric constant in air is equal to 1, the subsurfacelayer dielectric constant can be calculated from

?ε1 Am A0

Am A0

(2.2)

3

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where A0 and Am are the reflected amplitude from the top of surface layer and the metalplate, respectively. The dielectric constant of the subsequent layer is calculated from

?ε1 ?ε1

1

A0

Am

2 A1

Am

1

A0

Am

2 A1

Am

(2.3)

where A1 is the reflected amplitude from the top of subsequent layer. Moreover, a similarformula can be recursively obtained for the nth layer (Al-Qadi, 2004).

?εn ?εn1

1

A0

Am

2 n2

i1

γiAi

Am

An1

Am

1

A0

Am

2 n2

i1

γiAi

Am

An1

Am

(2.4)

where Ai is the reflected amplitude from the top of ith layer, and γi is the reflection coefficientbetween the ith and i 1th layers, i.e.

γi ?

εi ?εi1?εi ?εi1

. (2.5)

Since this technique relies greatly on the surface reflection, and because of the differentcompositions and ages of the layers in older pavements, the computed dielectric constantmay not be accurate. The error of thickness based on this dielectric constant estimation isaround 5 to 7.5% for asphalt layer thickness, and 9.5% for base layer thickness (David,2006). However, it was found in study that the dielectric constant decreases linearly withaging but remains within the typical limits (Al-Qadi, 2004).

Another approach to estimate the dielectric constant is based on the common depthpoint technique. Two sets of antenna were used simultaneously to estimate the average layerdielectric constant based on the collection of the response over the same location. The errorof thickness based on this dielectric constant estimation is around 6.8% compared to coring(Lahouar, 2002).

Water content

Since water has much higher value of dielectric constant than that of asphalt, investigationshave been done to track the relationship between the moisture content and dielectric con-stant. Topp et al. (1980) used a third-order polynomial to describe the relation between theeffective dielectric constant and the volumetric water content (VWC), where

εeff 3.03 9.3VWC 146.0VWC2 76.7VWC3. (2.6)

4

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Conversely, the measurement of εeff can be used to determine VWC by a third polynomialequation.

VWC 5.3 102 2.92 102εeff 5.5 104ε2eff 4.3 106ε3

eff (2.7)

More specific models can be found from the mixture theory, where the effective dielectricconstant of a pavement layer is a function of the dielectric constants of its constituents. Thesimplest mixture theory is the complex refractive index model, as described in Maser andScullion (1992). If asphalt layer is assumed to composed of air voids, aggregate, asphaltbinder, and water, its bulk (i.e. effective) dielectric constant is given by

?εeff Vair

?εair Vagg

?εagg Vbinder

?εbinder Vwater

?εwater. (2.8)

Another simple model suggested by Herkelrath et al. (1991) uses the following equation

VWC a?

εeff b, (2.9)

where a ¡ 0.These models suggest a monotonically increasing relationship between the effective di-

electric constant and moisture content of the mixture. It was found, however, that surfacewater on the pavement hardly affects the GPR data, unless the pavement structure is fullsaturated (David, 2006).

Density

Another material parameter affecting the dielectric constant is the mass density. The dielec-tric constant increases with increasing value of the layer density. Therefore this relation canbe utilized to detect stripping and air void content in pavement systems.

Stripping happens where the asphalt and aggregate bond fails, which leaves a weak layer.After continuously overlaying above such stripped layers that have lower density, surfacecrack often reappears within a short period of time. Since stripping indicates a layer with lowdensity, a negative reflection will be generated from the interface between layers. The GPRwas reported to be successful in detecting the location and extent of subsurface strippingat moderate or advanced stage (Scullion et al. 1996), although the presented approach isapproximate and could potentially be improved via the full waveform analysis.

Measuring air voids content using its dielectric constant is based on the fact that voidratio affects the overall density of the layer. Suitable empirical correlations can be obtainedto monitor the quality of new HMA surfacing (Saarenketo and Roimela, 1998).

5

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2.2 GPR waveform analysis

Integral representation

The waveform analysis approach offers the potential of better accuracy but requires a deeperknowledge of electromagnetic wave propagation. Since the entire electromagnetic response isavailable for backcalculation, this approach certainly has access to more information, whichpermits a better interpretation. To find the solution in layered structure, the key is touse the so called method of moments (MOM), where the response of the pavement system“illuminated” by electromagnetic waves is expressed in terms of integral representations.There are in general two alternative formulations: the electric field integral equation (EFIE)(Yu,2006), defined as

Eprq »V

GEpr, r1qJpr1qdr1

Hprq »V

GHpr, r1qJpr1qdr1(2.10)

where GH and GE are dyadic Green’s functions for the electric field of the medium. Theyare related to each other via

GHpr, r1q 1

iωµ∇ GEpr, r1q (2.11)

More precisely, the dyadic Green’s function GE is defined by the scalar Green’s functiongpr1 rq of the medium.

GEpr1, rq iωµ

I ∇∇

k2

gpr r1q (2.12)

where for an unbounded medium

gpr r1q eik|rr1|4π|r r1| , (2.13)

and the mixed potential integral equation (MPIE) (Yu,2006), defined as

Eprq iωAprq ∇φprq Aprq »V

GApr, r1qJpr1qdr1

Hprq 1

µ∇Aprq φprq

»V

GV pr, r1qρpr1qdr1(2.14)

6

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where Aprq is the magnetic vector potential due to the current density Jprq, φprq is thecorresponding scalar potential related to Aprq through the Lorenz gauge condition, GA isthe dyadic green’s function represents the magnetic vector potential due to current dipole,and GV is the scalar green’s function represents the scalar potential due to point charge.

GA pxGxxA zGzx

A qx pyGyyA zGzy

A qy zGzzA z (2.15)

where GijA is the green’s function in i direction due to current dipole in j direction, in matrix

form

GA

Gxx

A 0 0

0 GyyA 0

GzxA Gzy

A GzzA

(2.16)

The advantage of EFIE is that it makes it possible to solve problem of multiple burieddielectric and conducting objects. In addition, it allows one to compute the scattered filedand induced currents on the conducting surface. The downside of this method, however,is that GE and GH have higher-order singularities compared GA and GV in MPIE whichhave lower-order singularities and faster convergence. However, MPIE can handle only per-fectly conducting objects and only gives the induced currents on the conducting surfaces(Cui, 1999). Moreover, since the scalar potential kernel GV in MPIE has different form forhorizontal and vertical current components, it is not easy to extend the solution to generalnonplanar conductors (Michalski, 1997).

A combination of the above two methods is called the combined-field integral equation(CFIE), defined as

CFIE αEFIE p1 αqMFIE (2.17)

The improvement can be seen as it generally results in better-conditioned matrix equationand no internal resonance. Unfortunately, this method is not applicable for solely opensurface problems when compared to EFIE (Ergul, 2005).

Evaluation of the Green’s function

In the context of the above formulations, the major problem is how to efficiently evaluatethe Green’s functions in the integrals of (2.10) and (2.14). Several methods have beeninvestigated so far and will be illustrated below.

When the source and receiver are in the same layer and frequency belongs to somelimited range, one way to evaluate the green’s function is the complex image method (CIM).It basically expands the kernel via a series of complex exponentials as

G N

m1

amebmkzi . (2.18)

7

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Then the Sommerfeld identity is applied to obtain the closed-form solution in the form of

G N

m1

ameikirm

rm

direct term surface waves. (2.19)

The above expansion, however, becomes difficult to compute when the frequency is high orthe medium is multilayered as it is the case in pavement systems.

To expedite the integral evaluation, one can revise the integral path to such as steepestdecent path (SDP) and revised original path (ROP) (Cui,1999). Note that the SDP isefficient, but only valid in simple structures.

Recently, a new algorithm called fast interpolation and filtering algorithm (FIFA) (Yu,2006) is introduced. It is fast algorithm since it introduces a window function to force afaster decay of the integrand. However, this very window function introduces an error whichresults in low accuracy when the source is close to receiver.

Compared to the above developments, a better way to evaluate the Green’s function isbased on the method of asymptotic decomposition (Pak & Guzina, 2002), where the Green’sfunction in the transformed domain is decomposed into the asymptotic term and the residualterm. The asymptotic term is evaluated in a closed-form via analytical functions while theresidual term is evaluated numerically. The advantage of this decomposition is that sincethe integration path is from 0 to infinity, the singularity of the total kernel at infinity iscaptured by the asymptotic term as shown in Figure 2.1.

0 10 20 30 40 50 60 70 80 90 100−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ξ

kern

el

Ga

G

Gr

Figure 2.1: Asymptotic term Ga, residual term Gr and total kernel G

By taking out the singularity whose integration is available via algebraic function, the

8

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entire computation time is solely determined by the integration of the residual term. Hencethe evaluation speed can be greatly improved as well as accuracy. More precisely, one has

G 8»0

GaξJnpξρqdξ 8»0

pG GaqξJnpξρqdξ

Ga 8»0

GrξJnpξρqdξ,

(2.20)

where Gr G Ga is the residual term, and Ga is the asymptotic term whose integrationis in closed-form as Ga.

2.3 Summary

In summary, the dielectric constant can be used as an indicator to monitor physical quantitiesand features such as moisture content, air voids and stripping. The estimation of dielectricconstant is a primary factor affecting the accuracy of the layer thickness determination.Although the conventional (i.e. travel-time) GPR data interpretation shows the effectivenessfor certain occasions, the new technique of full waveform analysis is more versatile owing toits full exploitation of the measurement.

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Chapter 3

Correlation with Dielectric

In Chapter 2, it is pointed out that since the variation in the fractional volume of sub-stances can significantly affect bulk dielectric constants, the change of dielectric constants isassociated with the change of electrical properties, reflecting the change of the material orstructure. To describe such a change, it is worth investigating the factors affecting the valueof the dielectric constant.

Dielectric mixing models

There are two ways to determine the dielectric constant: (1) measured values and (2) mixingmodels. The latter one can provide an estimate of the dielectric constant for a bulk materialbased on specific parameters using (i) an empirical expression fulfilling the experimentaldata (Topp et al. 1980) or (ii) a semi-theoretical expression including volume fractions anddielectric constant of each constituent (Roth et al. 1990).

Topp et al. (1980) used a third order polynomial to describe the relation between effectivedielectric constant and volumetric water content (VWC):

εeff 3.03 9.3V WC 146.0V WC2 76.7V WC3 (3.1)

Conversely the measurement of εeff can be used to determine VWC by a third polynomialequation:

V WC 5.3 102 2.92 102εeff 5.5 104ε2eff 4.3 106ε3

eff (3.2)

More generally, the polynomial model is written as

V WC AB εeff C ε2eff C ε3

eff (3.3)

Another model suggested by Roth et al. (1990) uses the following equation:

εeff p1 ηqεns pη V WCq εn

a V WC εnw

1n(3.4)

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where η is soil porosity, and subscripts a, s, w represent air, soil and water respectively, andn is a factor accounting for the orientation of the electrical field with respect to the geometryof the medium. Note that n 0.5 is used for isotropic medium (Birchak, 1974), hence asimple physical interpretation of water content and ε was suggested by Herkelrath el. al.(1991):

V WC a?

εeff b (3.5)

Similarly, Al-Qadi et al. (2004) provided an expression to calculate the voids content inan asphalt sample by knowing parameters of dielectric constant and fractional volume of theaggregate and binder:

?εHMA Vair

?εair Vagg

?εagg Vb

?εb VW

?εW (3.6)

whereεHMA = dielectric constant of the hot mix asphalt layerVair = fractional volume of airεair = dielectric constant of the air, equal to 1Vagg = fractional volume of aggregateεagg = dielectric constant of the aggregateVb = fractional volume of asphalt binderεb = dielectric constant of the asphalt binderVW = fractional volume of waterεW = dielectric constant of the water, equal to 81.

A more simple expression based on (3.6) is one in which the dielectric constant onlydepends on air voids: ?

εHMA Va

?εair (3.7)

Conversely, the dielectric constant is the only dependent variable of air voids:

Va c?

εeff d (3.8)

3.1 Experimental procedure

Four sets of data were obtained from soil (fine-grained and coarse-grained soil samples), andfrom asphalt concrete (field samples and laboratory samples). All measurements of dielectricconstant were taken by percometer.

Percometer

The percometer (Figure 3.1), which consists of a 6-cm-diameter probe attached to a smallmicroprocessor, is an accurate and easy-to-use instrument. The measured value is the realpart of the complex relative dielectric constant. When the surface of the probe is pressedagainst the material to be tested, the device emits a small electrical current. Dielectric

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content (see Equation (3.5)) or air voids content (see Equation (3.8)) and effective dielectricconstant.

Table 3.3: Dielectric Constants

Material Dielectric Constant

Air 1

Asphalt 3 to 6

Clay 5 to 40

Concrete 6 to 11

Dry Sand 3 to 5

Granite 4 to 6

Limestone 4 to 8

Saturated Sand 20 to 30

Water 81

Bulk specific gravity of mixture Gmb

One important parameter in designing concrete mixes is specific gravity (Sp. Gr.), defined asthe mass of a material divided by the mass of an equal volume of distilled water. Four typesof specific gravity are defined based on how voids in the compacted mixture are considered.Three of these are widely accepted and used in portland cement and asphalt concrete mixdesign: dry weight, saturated surface dry, and apparent specific gravity. These are defined as

Dry Sp. Gr. Dry weight

pTotal particle volumeqγw

WspVs Vi Vpqγw

SSD Sp. Gr. SSD weight

pTotal particle volumeqγw

Ws WppVs Vi Vpqγw

App. Sp. Gr. Dry weight

pVolume not accessible to waterqγw

WspVs Viqγw

(3.9)

where

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Ws = weight of solidsVs = volume of solidsVi = volume of water impermeable voidsVp = volume of water permeable voidsWp = weight of water in the permeable voids in the SSD conditionγw = unit weight of water

Permeable and impermeable voids are illustrated in Figure 3.2. Only the permeable voidsare able to hold water. The amount of water the mixture absorbed is important in the designof portland cement. Depending on how moisture exists in the voids, there are three kind ofstates: bone dry, air dry, and saturated surface dry (SSD). There is no specific level of mixabsorption for a desired mixture, but mix absorption must be evaluated to determine theappropriate amount of water added to the mix.

Permeable Void

Impermeable Void

(a) bone dry

Moisture

(b) air dry

Moisture

(c) saturated surface dry

Figure 3.2: Voids and moisture absorption of mixtures

In this study, to simplify the problem, we assume that all voids are water permeablevoids. In this way, air voids can be calculated by dry specific gravity.

Maximum specific gravity of mixture Gmm

In designing a paving mixture with a given aggregate, the maximum specific gravity, Gmm,of a given asphalt content is needed to calculate the percentage of air voids. While themaximum specific gravity can be determined for each asphalt content by ASTM D 2041 orAASHTO T 209, the precision of the test is best when the mixture is close to the designasphalt content. Also, it is preferable to measure the maximum specific gravity in duplicateor triplicate.

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Percent air voids Va

According to ASTM D 2041 or AASHTO T 209, the percent air voids in a pavement samplecan be calculated using

Va 100 Gmm Gmb

Gmm

% (3.10)

whereVa = air voids in compacted mixture, percent of total volume

Gmm = maximum specific gravity of paving mixtureGmb = bulk specific gravity of compacted mixture

Since Gmb represents dry specific gravity, the calculated percent air voids is the total voidsin the compacted mixture.

Volumetric water content V WC

Due to the permeable voids, the mixture is able to hold a certain amount of water. Thecapability to hold water is not simply linearly related to the voids content, but it dependson the versatility of the compound structure and the absorption state. In this study, theSSD state and no impermeable voids are assumed. The volumetric water content (VWC) isdefined as

VWC SSD weight - Dry weight

Dry weight(3.11)

It can be also defined in terms of gravimetric water content (GWC):

VWC γm

γw

GWC (3.12)

where γm, γw are the unit weight of mixture and water respectively, and GWC is defined as

GWC SSD weight - Dry weight

SSD weight(3.13)

3.1.3 Data from laboratory asphalt concrete sample

The sample cores were prepared by gyratory compaction as 150mm in diameter and 180mmthick cylinders. Both limestone and granite aggregate were used, as well as two different airvoids contents, 4% and 7%.

The samples were first weighed before being submerged into water, and this weight wasapproximated to be the dry weight. After submersion into water for 15 minutes, the sampleswere removed and the surfaces were wiped dry; the samples were weighted again under theassumed saturated surface dry (SSD) condition. The effect of water on dielectric constantwas illustrated by placing the samples in the air and weighing every consecutive hour forthree hours, in which case the water content was gradually reduced.

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3.2 Summary

The ultimate goal in these experiments is to find the correlation between dielectric constantand material properties such as water content and air voids content, using the describedmodels by equations (3.3), (3.5) and (3.8). In addition, a linear expression is assumed forthe purpose of comparison.

VWC aεeff b, Va cεeff d (3.14)

3.2.1 Data from fine and coarse-grained soil

Fine-grained soil

The data set is shown in Appendix Table A-3 and Table A-4. As discussed before, the mixingmodels include linear (3.5) and polynomial (3.3) relations. Using (3.3), the polynomialtrendline is plotted in Figure 3.3 together with linear trendline (see Equation (3.14)). As

Figure 3.3: Fine-grained soil: dielectric constant vs VWC

shown in Figure 3.3, since the polynomial model only has a slightly higher R2 compared tolinear model, the linear model would be more convenient to predict VWC.

In Figure 3.4, the trendline of square root dielectric constant was plotted as expressed inequation (3.5). The value R2 0.5464 is close to the previous models.

Therefore none of the models are working for these fine-grained soils. It is possiblebecause of very small voids and a none-uniformly distributed water content.

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Figure 3.4: Fine-grained soil: square root dielectric constant vs VWC

Coarse-grained soil

The data set is shown in Appendix Table A-5. Since only gravimetric water content datawere provided, it is necessary to convert GWC into VWC by (3.13). Both polynomial andlinear trendlines were plotted in Figure 3.5.

Figure 3.5 illustrates that both linear and polynomial trendlines have similar R2 values.For simplicity, the linear one may be easier to use.

Using (3.5), the trendline of square root dielectric constant with R2 0.8576 is shownin Figure 3.6. Again, the R value is not much better than linear model.

3.2.2 Field data from MnDOT

According to the data in Appendix Table A-1, those missing maximum specific gravity andthose with negative values are unusable. After eliminating that data, the relation betweenVa/VWC and ε is readily obtained. According to equations (3.5) and (3.14), the trendlinesbetween VWC and dielectric constant are plotted in Figure 3.7 and Figure 3.8.

As shown in Figure 3.7 and Figure 3.8, both trendlines have very small R2, meaning thatthere is virtually no correlation between VWC and dielectric constant for these data.

In Figure 3.10 and Figure 3.9, air voids content and dielectric constant are plotted using(3.8) and (3.14).

As shown in Figure 3.10, the R2 is very small, indicating that there is no correlationbetween the air voids content and dielectric constant, as well as the square root of dielectricconstant. As can be seen, it is not possible to correlate dielectric constant and water contentfor this data.

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Figure 3.5: Coarse-grained: dielectric constant vs VWC

Figure 3.6: Coarse-grained: square root dielectric constant vs VWC

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Figure 3.7: Field asphalt concrete: square root dielectric constant vs VWC

Figure 3.8: Field asphalt concrete: dielectric constant vs VWC

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Figure 3.9: Field asphalt concrete: square root dielectric constant vs air voids

Figure 3.10: Field asphalt concrete: dielectric constant vs air voids

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One possible reason for such apparent lack of correlation is the local (near-surface) vari-ation of moisture in asphalt concrete specimens due to ambient humidity conditions. By itsnature, such boundary layer could have significantly altered the percometer measurementsdue to high value of the dielectric constant in water relative to that in asphalt (See Table 3.3),while affecting the overall VWC only minimally. One of the recommendations of this studyis to explore such possibility in future projects by either i) attempting to measure the di-electric of asphalt concrete more globally (i.e. using a device alternative to percometer thatpenetrates deeper into the material), or ii) performing dielectric measurements in an equi-librated moisture environment that would minimize the presence of the above-hypothesizedboundary layer.

3.2.3 Data from laboratory asphalt concrete samples

The lab asphalt samples were prepared by gyratory compaction with two different kindsof air voids, 4% and 7% (Appendix Table A-2). Limited by the available data and by themoisture inhomogeneity of the bulk as examined earlier, only qualitative observations areprovided.

• For the same sample, the value of dielectric constant of the dry state was lower thanany other state (with different drying times).

• For the same sample, the longer the drying time, the smaller the value of the dielectricconstant.

• The 4% air voids had, an average, a higher value of dielectric constant than that of 7%air voids at the dry state, and the trend was opposite at the saturated state.

• With the same air voids content, the limestone aggregate sample tended to have ahigher effective value of dielectric constant than that of the granite aggregate.

Thus the value of the dielectric constant seems to be increase with the increase of watercontent. In particular, increasing drying times result in diminishing residual water contents,which in turn produce decreasing values of the dielectric constant. Since the value of dielectricconstant in air is smaller than the limestone or granite aggregate (see Table 3.3), the 4%air voids sample would have a higher effective value than that of 7% air voids sample. Alsobecause water has a much higher value of dielectric constant, the saturated 4% air voidssample would possess less water and a smaller effective value of dielectric constant as well.Therefore water content has a great potential to affect the dielectric constant.

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Chapter 4

Electromagnetic Waveform Analysisin Layered System

As discussed in Chapter 2, the benefit of using full-waveform analysis is to exploit the GPRdata as much as possible, which increases the opportunity to obtain better interpretationof layer properties. Because the full-waveform analysis for layered system is the foundationof this project, it deserves a single chapter for future reference even though the elaboratederivation is given in the Appendix B.

The electromagnetic field is described by the Maxwell’s equations. The interrelationshipbetween electric field (E), magnetic field (B), electric charge density(%), and electric currentdensity (J) are formulated by a set of four equations, where, in vacuum, the Maxwell’sequations are given as

∇ E %

ε0

∇ B 0

∇ E BBB t∇B µ0J µ0ε0

BEB t

(4.1)

where ε0 8.85419 1012F/m as the permittivity in vacuum and µ0 4π 107H/m asthe permeability in vacuum. In terms of the speed of light c, one has that

c 1?µ0ε0

. (4.2)

In linear medium, the free space dielectric constant ε0 is replaced by the material electricpermittivity ε, and the free space magnetic permeability µ0 is replaced by the materialmagnetic permeability µ as well. Therefore, Maxwell’s equations are available in any medium

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to describe the field strength via

∇ E %

ε∇ H 0

∇ E µBHB t

∇H J εBEB t

(4.3)

where H Bµ is introduced as auxiliary magnetic field.The solution of the Maxwell’s equations for unbounded system is available in the litera-

ture. In layered system such as the ideal pavement system, however, the solution of (4.3) isleft to be filled in this project. The schematic of a layered system is shown in Fig. 4.1, where

Figure 4.1: Schematic of layered structure

the source (transmitter) T and receiver R are putting in the 0th layer which is the upper

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half space (air). The cylindrical coordinate is set on the ground with positive z-directiondownward. The spacing information are shown in the figure: the horizontal separation ρbetween T and R, the height s for source and the height z for receiver. This setup is appliedin order to simulate the GPR configuration where both transmitter and receiver are placedin the box above the ground at the same elevation, i.e., s z. There are total n 2 layerslabeled from 0 (upper half space) to n 1 (lower half space). Each layer is characterizedby its properties, namely the dielectric constant εj, permeability µj, conductivity σj, andthe layer thickness hj zj zj1 pj 0, 1, , n 1q, where zj is the depth at the bottomof the jth layer. Note that z1 and zn1 represent the artificial outside boundaries of thetwo half spaces, where the infinity layer thickness should be imposed on. For computationalpurpose, however, a large preset value is enough to indicate the infinity thickness.

Upon placing the source T in the lth layer, the EM wave would propagate through all thelayers. The received EM field at receiver R is the wave field after the multiple reflections andtransmissions on all the layer interfaces. The source and receiver are in general arbitrarilylocated in the layered system. In this project, however, the source T and receiver R happento be at the same elevation in the top layer (air) due to the configuration of GPR system.The elaborate solution is shown in the Appendix B.

As shown in the Appendix B, the forward model is capable of fast evaluation of the EMfield in layered system, which permits the reproduction of the field data from GPR survey.Furthermore, due to the fast evaluation, it is feasible to generate a data base in order tobackcalculate the pavement parameters.

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Chapter 5

Backcalculation Algorithm

Based on the forward model discussed in the Chapter 4, it is possible to produce the syntheticGPR measurements, i.e. the full electromagnetic waveform (GPR scan) with the time markand amplitude of the pulse. These waveforms, which contain the information about the layerthickness and dielectric constant, are individually different. In order to assist the analysisin pavement management, those pavement properties should be backcalculated from themeasured data. Unlike the traditional travel-time technique, the measurement should beinterpreted in a full-waveform fashion which has the advantage of exploiting a lot moreinformation from the available data. The focus of this chapter therefore is to introduce afull-waveform back-analysis numerical scheme, which is based on the artificial neural network(ANN) algorithm and frequency response function (FRF) concept.

5.1 Artificial neural network (ANN)

Introduction to ANN

Generally speaking, the artificial neural network (ANN) is introduced as a computationaltool to simulate biological neural networks in neuroscience. In practice, it can be used asdata modeling aid to simulate complex relationships (patterns) between the input and theoutput. The type of neural network used here is called feed-forward network, in which theinformation is transmitted in only one direction, from the input nodes to the output nodes,with no cycles or loops.

Computing unit

Each node, also called neuron, is a computing unit, which performs certain function. Oneexample of an abstract node is shown in Fig. 5.1. The function g on the left side of the nodeis called the integration function, which generates excitation gpxq from the prescribed inputx. The function f on the right side of the node is called the activation function, which usesexcitation gpxq to generate the output y fpgpxqq. The g function, in general, is an addition

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function, whereas the function f employed here can be either linear function or tan-sigmoidfunction (see Fig. 5.2).

The nonlinear tan-sigmoid activation function permits the network to nonlinearly mapthe input from 8 to 8, to an output from 1 to 1. From Fig. 5.2(b) one can seethat the tan-sigmoid function becomes more like a step function by increasing the value c.Note that c 1 is applied in this project. On the other hand, the linear activation functionprovides a convenient tool to map the input to a value outside the range between 1. As aresult, a network combining these two kinds of nodes is capable of having arbitrary outputvalues.

Network architecture

A feed-forward network can be constructed with either one or multiple layers. Multi-layernetwork can be constructed in the similar fashion as the one-layer network. Usually, thereare two more layers constructed for input and output alone. As an example, a commonlyused three-layer feed-forward neural network, which is the simplest, is shown in Fig. 5.3.

On the architecture level, the example network in Fig. 5.3 contains the input layer, thehidden layer and the output layer arranged in the sequential order. Each layer consists of anumber of nodes. All the nodes except those in the input layer are computing units. Everynode connects only to the nodes in other layers. No connections are allowed between nodeswithin one layer. In each layer, one bias node is connected to each node in the layer, exceptin the input layer. The bias is much like a weight, except that it has a constant input of 1.

The number of nodes determines the complexity of the model. In nature, more compli-cated relationship can be archived with more nodes inside the network. Usually, the numberof nodes in the input layer equals to the number of inputs (I); the number of nodes in theoutput layer equals to the number of outputs (O). The number of nodes in the hidden layer(H) can be adjusted to satisfy the particular requirements.

By adding weight to each connection between two nodes, the network is engaged withmore possibilities to model the relationship between the input and the output. The weightrepresents the connection strength of the input and the weighted total sum of the input; it

Figure 5.1: Node–computing unit

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Figure 5.3: Three-layer feed-forward neural network

W2pf1pW1x b1qq b2. After all, the output for the overall network function is obtainedas in (5.1) recursively.

Training the network

Neural network is good at performing a particular function only if it learns to do so. Thenetwork can be trained so that a particular input leads to a specific target output. Oncethe network is defined such that the number of nodes and its architecture are fixed asshown in Fig. 5.3, the relation between the input and output is purely determined by theconnection between nodes, i.e., the weights and biases. Learning algorithm can be dividedinto supervised and unsupervised learning. In supervised training, which is used in thisproject, a set of input and desired target pairs are given. The input is sent to the network

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to obtain the network response. By altering the weights associated with the connection, theoptimum network function is obtained when the minimum error between the target outputand the network response (output) is met (see Fig. 5.4). The network function represents thepattern inside the training data set. The procedure to find the optimum network functionis called training. The set of input-target data pairs used to train the network is called thetraining data set.

Figure 5.4: Principle of training neural network

Plasticity of network

The number of degrees of freedom (number of weights) available to the network determineswhether or not the pattern inside the training set can be learned exactly. The plasticity ofthe network is determined by the number of degrees of freedom, which means the capabilityto approximate the pattern. Since the obtained pattern is useful only when it is valid tonew data as well as to given data, the trained network has to be verified with new data. Asa rule of thumb, too little plasticity leads to large error in both given and new data. Toomuch plasticity, however, results in less error in given data but also leads to large error innew data. In other words, more than enough plasticity hurts the generality of the networkas well as networks with less than enough plasticity. This is shown as an example in Fig. 5.5.

In this example, a noisy sine function is studied by a neural network. The dotted line isthe underling sine function, where the noisy sine signals are shown with a ’+’ sign. The noisydata are sent as training data set to a neural network that has too much plasticity (numberof nodes more than enough). The solid line is output obtained from such a network, in whichthe network tried to fit every single noisy data. Apparently, the network has overfitted thetraining data, which it failed to generalize well.

In conclusion, the generalization ability depends on the network’s plasticity, which inturn is a function of the number of hidden nodes. There is no rigorous rule to define whatis the sufficient number of nodes for the network. In the ideal case, the number of degreesof freedom of a network (number of weights between nodes) should not be more than thenumber of training data sets, because otherwise the network is in danger of being overtrained.

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Figure 5.5: Generality of function approximation on a noisy sine function

5.2 Frequency response function (FRF)

Introduction to FRF

For a constant-parameter linear system, which is the case of the GPR system, its dynamiccharacteristics can be described by frequency response function (FRF). An important prop-erty of such a system is that only the amplitude and phase of an applied input can bemodified by the system, but the frequency is preserved, which means there is no translationor ”leakage” of power between frequencies. If the system output, denoted as yptq, is due toan input xptq, the FRF characterizing the system is defined as:

FRFpfq Y pfqXpfq (5.3)

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where Xpfq and Y pfq are the Fourier transforms of the temporal records xptq and yptq,respectively defined as:

Xpfq 8»0

xptqei2πftdt

Y pfq 8»0

yptqei2πftdt.

(5.4)

It is noted that the lower integration limit is 0 since xptq yptq 0 when t 0.In the GPR system, because the full waveform is generated by the flowing current, the

applied current Iptq is the system input, where the GPR measurement Eptq is the systemoutput. After Fourier transformation to the frequency domain, the FRF of the GPR systemcan be defined as:

FRFGpfq EpfqIpfq (5.5)

Note that the FRF function is, in general, a complex value.

Estimation of FRF

In general, two ways are employed to calculate the FRF, (i) direct division in equation (5.3)and (ii) calculation of the relations between power spectra and cross-spectra.

(i) Since the collected measurements are discrete data point within finite time duration,assume that N points of data are evenly sampled at time interval ∆t. The signal xptq andyptq at t n∆t pn 0, 1, 2, , N 1q are replaced by discrete form.

xn xpn∆tq, yn ypn∆tq, n 0, 1, 2, , N 1 (5.6)

In addition, the Fourier transform at N discrete frequency values fk pk 0, 1, 2, , N 1qis computed, where

fk kf k

N∆t(5.7)

Note that, for computational purposes, the number of discrete frequency values is the sameas the number of sampled data. Then the discrete Fourier transform is obtained via

Xk Xpfkq∆t

, Yk Y pfkq∆t

, k 0, 1, 2, , N 1 (5.8)

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Thus, the discrete version of (5.4) are readily shown as

Xk N1

n0

xn exp

i2πkn

N

Yk N1

n0

yn exp

i2πkn

N

(5.9)

where k 0, 1, 2, , N1. Based on (5.3),(5.8) and (5.9), when i 1, 2, ,M realizationsof GPR tests are available, the FRF is calculated by

FRFpfkq 1

M

M

i1

Yk

Xk

i

. (5.10)

(ii)As described in Bendat and Piersol (1971), for a pair of sampling records xptq andyptq from stationary random process txptqu and typtqu, the FRF can be calculated as theratios of the cross-spectral density Sxy to the power spectra density Sxx, via

FRFpfq SxypfqSxxpfq (5.11)

where the spectral density functions are defined as

Sxypfq 1

TXpfqY pfq, Sxxpfq 1

TXpfqXpfq (5.12)

where Xpfq and Y pfq are the Fourier transforms of xptq and yptq.At this point, an important function called the coherence function should be introduced

as

γ2xypfq |Sxypfq|2

SxxpfqSyypfq . (5.13)

After replacing the spectral functions in (5.13) with (5.12), it follows that

γ2xypfq |XpfqY pfq|2T 2

|Xpfq|2|Y pfq|2T 2 1 (5.14)

This means, in ideal case of a constant-parameter linear system with no extraneous noise,the coherence function between xptq and yptq should be unity. Thus, the coherence functionindicates how well these two signals are related. For example, if xptq and yptq are completelyunrelated, the coherence function is zero. If the coherence function is between zero and one,possibilities are that (i) there is extraneous noise in the measurements in xptq, yptq or both,(ii) the system is not linear as it is assumed, or (iii) there may be another input other thanxptq which affects yptq. Thus, the quality of the measurement should be calculated before

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computing the FRF, where ”bad” measurements should be discarded.When multiple realizations (i 1, 2, M) of the GPR tests are available, analogous to

equation (5.7) and (5.8), the discrete form of (5.12) can be written as

Sxypfkq ∆t

MN

M

i1

pXk Ykqi

Sxxpfkq ∆t

MN

M

i1

pXk Xkqi

(5.15)

where N∆t T is the total duration of records. The FRF characterizing the system is thencomputed as

FRFpfkq SxypfkqSxxpfkq , k 0, 1, 2, , N 1 (5.16)

and the coherence function is given as

γ2xypfkq |Sxypfkq|2

SxxpfkqSyypfkq , k 0, 1, 2, , N 1 (5.17)

FRF patterns

Note that the FRF is generally a complex-valued quantity. As an example, Fig. 5.6 showsthe real and imaginary components of FRF, which are the functions of frequency. In aconstant-parameter linear system, the shape of the FRF curve is determined by the systemparameters that can in turn be identified by interpreting the pattern of FRF. As a note, dueto noise contamination during (e.g. GPR) survey, only the FRF within the range aroundoperating frequency is considered to have high S/N ratio. The outside range is noise-pollutedor attenuated, therefore is no good for identification.

Due to a digitized (e.g. discrete) nature of experiment data, only discrete data pointsare involved. To identify the pattern, the simplest way is to analyze the FRF values overthe frequency range of interest. In such scheme, however, it is not convenient to deal withindividual frequencies which can be highly affected by the experimental noise. As an alterna-tive, the FRF pattern is identified by calculating several segments’ area under the curve. Asan example illustrated in Fig. 5.6, within the frequency range of interest from f1 to f8, theFRF pattern is defined by a combination of seven evenly divided segments. Each segment isassociated with two areas under the real and imaginary components of FRF. For instance,the fourth segment as shown in Fig. 5.6, is associated with two areas, Rearea

4 and Imarea4 .

These segmental areas which constitute the FRF pattern are inserted as one set of inputsinto the neural network. The neural network automatically interprets the input FRF patternto the most similar target FRF pattern, if the set of the target parameters are the most similarset of the input parameters. In other words, the neural network is capable of identifying thesimilarities between the input patterns and the possible pavement parameters.

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Figure 5.6: Areas under FRF

5.3 Numerical scheme

GPR system profile

The neural network has to be trained before used to interpret the pavement properties,and the training data is generated by the forward model introduced in Chapter 4. Assumethat the GPR box is placed above the pavement with transmitter and receiver at the sameelevation. As shown in Fig. 5.7, the spacing information (z, d1, d2, ρ) is available, namely theheight z from the transmitter/receiver to the ground, and the spacings d1, d2 and ρ insidethe box.

Figure 5.7: Spacings of GPR antenna box

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For the two types of GPR antennas operated by Mn/DOT, namely 4105 (2GHz antenna)and 4108 (1GHz antenna), these spacing values are slightly different as listed in Table 5.1.In addition, the height z is fixed at 0.9144 m for both.

Table 5.1: GPR antenna box inside spacings

Type d1 [m] ρ [m] d2 [m]

4105 0.0762 0.381 0.0762

4108 0.1016 0.335 0.1016

Due to the fact that the GPR signal is highly attenuated below the base layer, there isvery weak reflection from the bottom of the base layer. Therefore, the pavement system isconsidered as a two-layer system: an asphalt surface layer and a base layer, where the baselayer is assumed to be an infinite half space. A schematic of the assumed GPR-pavementsystem is shown in Fig. 5.8. Three unknown pavement properties as the targeted estimationsare circled, namely the dielectric constants of asphalt layer εA and base layer εB, and thethickness of asphalt layer hA. The possible range of the featured pavement properties islisted in Table 5.2, where ε0 8.8542e12F/m is the dielectric constant in the air.

Table 5.2: Pavement properties with variable parameters

Dielectric Constant Thickness[F/m] [m]

Asphalt εA 4ε0 8ε0 hA 0.0508 0.254m

Base εB 6ε0 12ε0 hB 8

Backcalculation procedures

As stated earlier, using a properly-selected training data set, the neural network can betuned to determine the correlation between the input and output. This is accomplishedwith synthetic GPR data generated by a computer program, which implements the modelof electromagnetic wave propagation in a three dimensional layered pavement structure.

To train the neural network, pavement properties covering the possible range in Table 5.2will be inserted into the model, as well as the fixed parameters such as spacing z and ρ, airdielectric constant ε0, and operation frequency f . Despite the temporal shape of the inputsource, i.e. current pulse, the FRF of the linear system is preserved. Further, the smallestcomponent of any pulse antenna, i.e. the dipole source, is assumed in this study to generatethe electromagnetic field.

For the purpose of training, the FRF components in terms of the 14 segmental areas (seeFig.5.6) are computed as discussed before. They are sent to the neural network as input,

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Figure 5.8: GPR implementation configuration

where the target output is the variable pavement properties implemented in the forwardmodel. Note that the two categories of the training and the test data set should be generatedseparately, where the training data is used to train the neural network and test data is usedto verify the trained network.

As a starting point, the network with one hidden layer (I-H-O) is designed, where thenumber of nodes, I and O, are equal to the number of inputs and outputs (e.g. I 14,O 3), respectively. The maximum number of nodes Hm in the hidden layer is empiricallydetermined by

Hm floor

Neqr O

I O 1

(5.18)

where Neq is the number of independent training equations, i.e., the number of input-outputdata sets if there are no duplicates. The redundancy parameter r typically varies from 8to 20 if Neq is sufficiently large. This estimation of Hm is only for practical purposes. Thenetworks with number of hidden nodes larger than Hm are in danger of being overfitted.However, it is good practice to vary H from Hm in order to obtain the optimum relationshipbetween input and output.

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To test the generalization ability of alternative (competing) networks with various archi-tectures, neural networks with one output (O 1) in the output layer are also examined. Inthis case, the network is a combination of three single output networks, where each networkis trained separately for a different target.

Selection of parameters

As an illustration, the 1GHz antenna measurement is simulated by the forward model, wherethe frequency range of interest is taken as 0.8-1.2GHz. The parameters of pavement proper-ties implemented in the forward model are listed below.

1.For the fixed parameters: z 0.8763m, ρ 0.335m, ε0 8.8542e12F/m.2.For the variational parameters, a total of N random selections for each parameter is

specified, via

pεAqi

4 p8 4qR i

N

ε0

pεBqi

6 p12 6qR i

N

ε0

phAqi 0.0508 p0.254 0.0508qR i

N

(5.19)

where i 0, 1, , N 1 and R is random number from 0 to 1. For practice purposes, inorder to randomly spread the parameters (εA, εB and hA) over their own data spaces, itis important to generate R without duplication for every single entry. Therefore, the totalcombination of these three parameters is N3, e.g., for N 25, there are N3 15625 trainingdata sets.

Neural network results

It should be noted that all the neural networks in this study are implemented in MatlabNeural Network Toolbox (v5.1). This toolbox provides a simple routine to automaticallybuild networks with desired layers, nodes, connection weights and biases. Then the neuralnetwork is trained with randomly selected input-output data pairs (xtr, ytr) according to(5.19), where output ytr are the random parameters of pεAqi, phAqi and pεBqi and input xtr

are the system FRFs computed by the forward model assuming ytr and the remaining fixedparameters. New test data pairs (xts, yts) are generated separately. During the test stage,the network response ats is obtained due to the input xts. The linear regressions between ats

and the target yts are performed, and the correlation coefficients (R-value) are calculated aswell, where the higher R-value represents the better performance of the network.

In what follows, the neural network is designed to have 3 outputs simultaneously (O 3),as shown in Fig. 5.9. As guided by equation (5.18), we select Neq 15625, r 17, I 14and O 3, the Hm is calculated as 51. Therefore, the first neural network is created bysetting the number of hidden nodes at H 51. Then additional networks with different

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Figure 5.9: Network with three outputs

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number of hidden nodes are created by decreasing H of 5 each time until H 11, meaningthere are only 11 nodes in the hidden layer. In this way, the number of hidden nodes withrespect to optimum network function can be found by comparing the R-value. There aretotal of 1000 pairs of test data prepared. Each network are verified with the same test data,and the R-values of linear regressions are listed by the order of value H in Table 5.3.

Table 5.3: H vs R-values

H RpεAq RphAq RpεBq11 0.9996 0.7387 0.998816 0.9999 0.9146 0.999521 0.9999 0.9148 0.999526 0.9999 0.8781 0.999731 0.9999 0.9364 0.999636 0.9999 0.9210 0.999641 0.9999 0.9222 0.999646 0.9999 0.9161 0.999651 0.9999 0.9342 0.9996

From Table. 5.3, the maximum R-values for all three parameters (εA, hA and εB) are0.9999, 0.9364 and 0.9997, respectively, indicating the network is successful in predicting thepavement parameters, namely dielectric constant of asphalt and base layer (εA, εB) and thethickness of the asphalt layer (hA). The highest R-value of hA is 0.9364 when the number ofhidden nodes H is 31, whereas RpεAq 0.9999 and RpεBq 0.9996. Therefore the optimumvalue of H is around 31, where the linear regressions results are plotted in Fig. 5.10.

Another possible spelling for the neural network is a combination of three different single-output networks, as shown in Fig. 5.11. One network is created for each pavement propertysought. Because the single output network requires less degrees of freedom compared tomultiple outputs network, the size of each network is expected to be smaller. Therefore,Neq 153 3375, r 8 and O 1 are selected this time, where I 14 is kept the sameas before. As guided by equation (5.18) again, Hm 27 is calculated, which means that themaximum of 27 hidden nodes is needed in each network. Similarly, the network is createdfirst with H 27 hidden nodes. Then different networks are created by decreasing H by avalue of 3 each time until H 9. one thousand pairs of test data are generated to verifythe trained network with respect to different H values, and the R-values are calculated asshown in Table 5.4.

From Table. 5.4, the maximum R-values for all three parameters, εA, hA and εB are1.0000, 0.9978 and 0.9998, respectively, which means the network is successful in predictingthe pavement parameters of the dielectric constants of the asphalt and base layers (εA, εB)and the thickness of the asphalt layer (hA). The highest R-value of hA is 0.9978 when thenumber of hidden nodes H is 27, whereas RpεAq 1.0000 and RpεBq 0.9998. Thereforethe optimum network is obtained when H is around 27, where the linear regression results

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(a) R 0.9999 (b) R 0.9364

(c) R 0.9996

Figure 5.10: Linear regressions for (a) εA, (b) hA and (c) εB (H 31)

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Figure 5.11: Network combined with three single output networks

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Table 5.4: R-values vs H

H RpεAq RphAq RpεBq9 1.0000 0.9940 0.998812 1.0000 0.9935 0.999615 1.0000 0.9938 0.999718 1.0000 0.9965 0.999621 1.0000 0.9951 0.999824 1.0000 0.9973 0.999827 1.0000 0.9978 0.9998

of this strategy are plotted in Fig. 5.12.

Conclusion

The above develops two alternative architectures of the neural network, a multi-outputnetwork with highest RpεAq 0.9999, RphAq 0.9364 and RpεBq 0.9996 vs single-outputnetwork with highest RpεAq 1.0000, RphAq 0.9978 and RpεBq 0.9998, the performanceof the latter one slightly surpasses the former one. When putting in the consideration thatthe larger size of the network requires larger memory storage and more computation time,it is recommended to use the combination of three single-output networks since this kindof network has smaller architecture than that of multi-output network. In addition, thepreliminary results from neural network are obtained via noise free synthetic data whichshould be modified in accordance with the ambient noise during the GPR survey. In otherwords, the noise with certain level has to be injected for the training data in order to increasethe network’s flexibility, which obviously increases the required number of the hidden layer.As a result, the more noise injected, the larger the benefit of the combined network over thesingle network in terms of the memory storage and computation time.

5.4 Single vs multiple sublayers in asphalt

Simulation with two-layer structure

The presumed layered system only includes two layers, namely the asphalt layer and baselayer (see Fig. 5.13(a)). Occasionally however, there are multiple “lifts” or individual layersmaking up the total asphalt thickness which means the first layer consists of more than onlyone layer (see Fig. 5.13(b)).

The question arises what will be the overall asphalt layer properties shown in Fig. 5.13(b)that can be captured by the backcalculation scheme for the configuration in Fig. 5.13(a)?Either the effective layer properties of all three overlays in asphalt or only the layer propertiesof the top overlay will be considered.

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(a) R 1.0000 (b) R 0.9978

(c) R 0.9998

Figure 5.12: Linear regressions for (a) εA, (b) hA and (c) εB (H 27)

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(a) Asphalt of a whole layer (b) Asphalt of three different layers

Figure 5.13: Whole layer vs different layers in asphalt

The selected dielectric properties of the three different overlays are listed in terms of therelative dielectric constant

εA1 6.1ε0, εA2 5.2ε0, εA3 4.5ε0 (5.20)

where hA1 hA2 hA3 0.8 in. By applying the backcalculation scheme which assumesthe existence of only one layer in asphalt (see Fig. 5.13(a)) to the configuration containingoverlays shown in Fig. 5.13(b), the determined asphalt layer parameters are the following

εA 6.27ε0, hA 3.87in. (5.21)

As a result, the effective asphalt dielectric constant (6.27ε0) is close to the dielectricconstant of the top overlay (6.1ε0), whereas the computed total thickness is deviated fromthe real one. First of all, the top overlay would reflect a large portion of the total energy,which easily results in an overall determined dielectric constant close to that of the topoverlay. On the other hand, after the pulse propagates through the surface layer, there isstill energy reflected back from the interfaces of the other layers which is not consideredin the backcalculation scheme for a three-layered system. Consequently, the total energyis disturbed, which leads to a layer thickness error as determined by the backcalculationscheme.

Besides the case of multiple thickness layers, the pavement usually contains thin layerswhich have different dielectric values than that of full-depth composite layer. To study thethin layer effect, the results of a two-overlay structure which has a thin layer over a normallayer inside the asphalt are presented here.

First we examine a three-layered structure which is presented here as a reference. The

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pre-defined layer properties are the following:

Case 1 : ε r1 8 12sε0, h r8 5.58sin. (5.22)

which means that layer one has a dielectric constant of ε0 and the layer thickness is infinitysince it is the upper half space (air); layer two has a dielectric constant of 8ε0 and layerthickness of 5.5 in. (asphalt layer); layer three has a dielectric constant of 12ε0 and the layerthickness is considered to be infinity since it is the lower half space (base layer).

By applying the backcalculation scheme, the determined effective asphalt layer propertiesare that

Case 1 : ε r1 7.98 11.69sε0, h r8 5.278sin. (5.23)

which are very close to the true values as expected.The second example has one more thin layer (h=0.5 in.) with smaller dielectric constant

inserted on the top of the asphalt layer while the total layer thickness of the asphalt is keptthe same, i.e.,

Case 2 : ε r1 2 8 12sε0, h r8 0.5 58sin. (5.24)

In return, the determined layer properties are the following.

Case 2 : ε r1 4.8 12.34sε0, h r8 3.328sin. (5.25)

As one can see, because of the thin layer which has lower dielectric constant value, the overalldielectric constant is reduced from the normal value towards the smaller value, whereas thedielectric value of base layer has little change. However, the determined total layer thicknessof asphalt (3.32 in.) is deviated from the true value (5.5 in.).

In the third and fourth examples, the thin layer is still on the top of the asphalt layerbut with higher dielectric constant values than that of the normal layer underneath, i.e.,

Case 3 : ε r1 8 2 12s, h r8 0.5 9.58sin.Case 4 : ε r1 8 2 12s, h r8 0.5 78sin.

(5.26)

As a return, the determined layer properties are that

Case 3 : ε r1 2.5 11.58s, h r8 10.128sin.Case 4 : ε r1 2.1 15.25s, h r8 5.368sin.

(5.27)

In both case 3 and case 4, the determined dielectric constants of asphalt (2.5ε0 and 2.1ε0)are close to the values of the normal layer (2ε0), whereas the determined dielectric value ofbase layer in case 3 (11.58ε0) is close to the true value (12ε0) and in case 4 (15.25ε0) it is not.For the layer thickness, in case 3, the determined total thickness of asphalt is 10.12 in. whichis very close to the true value (10 in.), whereas in case 4, the determined total thickness ofasphalt (5.36 in.) is not close to the true value (7.5 in.).

In conclusion, because the thin layer is very small, smaller than the wavelength, the

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electromagnetic wave will propagate through the whole layer without much change. However,the first reflection from the top of the thin layer will still has effect to some degree. Thus,the overall dielectric constant tend to be deviated from the value of normal layer towardsthe value of the thin layer. In certain occasions, the thin layer has almost none effect to thewave propagation, therefore, the determined layer thickness is close to the actual thicknessas shown in case 3.

Simulation with three-layer structure

To address the problem with sublayers inside the asphalt, the more practical way is to redefinethe layer structure for the forward model and neural network. Instead of assuming a two-layer pavement system (asphalt and base layer), a three-layer pavement system includingtwo sublayers and one base layer should be employed as shown in Fig. 5.14, where the totalnumber of parameters has increased to 5. Therefore new simulations of GPR data has to begenerated, and new neural network has to be trained as well.

Figure 5.14: Two sublayers in asphalt

To reduce the time to generate training data for the neural network, two sublayers havebeen assumed with equal thickness, with different dielectric constants. In this newly de-fined system the number of unknown parameters is 4, namely the dielectric constant of thefirst/second sublayer in the asphalt (εa,1 and εa,2), the dielectric constant of the base layer(εb), and the thickness of the first sublayer (ha,1). Even with this reduced configuration, withten different values for each parameters the total number of the data sets is 104 10, 000. Inaddition, based on the experience of the backcalculation with neural network, the dielectricconstant in the first sublayer always has the better R value than the other parameters. Asa result, one may decrease the needed number of different values for εa,1 and increase that

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for ha,1. The modified scheme to generate the training data set has 5 values for εa,1, εb and20 values for εa,2 and ha,1, which results in a total number of 10,000 data sets.

In the training stage, different architectures of neural network including single networkwith 4 outputs and combination of 4 networks with 1 output of each one is applied, andvarious number of hidden nodes from 10-40 is applied as well. Based on a large number ofsimulations, it turns out that the combined neural network with H 30 hidden nodes ineach one has the best performance. The detailed results for each network are omitted herefor simplicity.

To show the effectiveness of the new configuration, numbers of circumstances with dif-ferent layer parameters has been verified. For instance, assume ε r1 4 8 10sε0 andh r8 1.5 1.5 8s in., the determined parameters are ε r1 3.99 8.35 9.71sε0

and h r8 1.48 1.48 8s in. A selected list of the results of test runs are shown inTable 5.5.

As shown, the error of the determined dielectric constant and thickness of the first sub-layer is quite small, as low as 0.089%, whereas the errors of the determined dielectric constantof the second sublayer and base are acceptable for some instances. Also, it is observed thatin the case where the two sublayers have equal (or close) dielectric constant of small value( 4ε0), the error of the determined ha,1 is larger compare to the case where the two sublayershave different dielectric value. Fortunately, unless severe stripping happens, the circumstancewith small dielectric value through out the whole asphalt layer is very limited. Therefore,one may determine the parameters of the structure with two sublayers and one base layerby the back-analysis scheme introduced in this chapter.

5.5 Summary

The backcalculation scheme is proposed and verified with synthetic data generated basedon the EM model introduced in Chapter 4. The results show that the scheme works wellfor interpreting the parameters of pavement system. However, it should be noted thatdifferent layer structures require a different setup for the scheme. For example, if sublayersare assumed in the asphalt, at least the three-layer system instead of two-layer system hasto be applied. Otherwise, it is beyond the capabilities of the backcalculation. Given enoughtime to generate data and train the network, this backcalculation scheme will work as longas the assumed structure fits the desired system. In addition, several notes about the neuralnetwork are listed below for future reference.

First, the neural networks developed in this study are all implemented using MatlabNeural Network Toolbox (v5.1). They may not be back-compatible with previous Matlabversions.

Second, the most effective architecture of neural network was found to be the three-layerfeed-forward network. More than one hidden layer is not necessary unless there are limitedcomputing resources, such as available memory. Because the nodal information has to bestored in a computer when computation is performed for a layer, large number of nodes inone layer directly results in large memory storage needs. Since the plasticity of network is

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Table 5.5: Test runs of 3-layered system: 2 sublayers + base layer

Defined value Determined value

ε r1 4 8 10sε0 ε r1 3.99 8.35 9.71sε0

h r8 1.5 1.58s in. h r8 1.48 1.488s in.

ε r1 4 8 10sε0 ε r1 3.98 7.41 9.30sε0

h r8 2.5 2.58s in. h r8 2.36 2.368s in.

ε r1 4 8 10sε0 ε r1 3.96 8.25 9.32sε0

h r8 3 38s in. h r8 2.9 2.98s in.

ε r1 4 8 10sε0 ε r1 3.97 8.08 9.68sε0

h r8 4.5 4.58s in. h r8 4.54 4.548s in.

ε r1 6 3 15sε0 ε r1 6.09 2.35 14.75sε0

h r8 1.1 1.18s in. h r8 0.97 0.978s in.

ε r1 6.5 4.2 12sε0 ε r1 6.58 4.53 12.80sε0

h r8 1.8 1.88s in. h r8 1.67 1.678s in.

ε r1 8.2 2.2 16sε0 ε r1 8.30 2.18 15.76sε0

h r8 2.5 2.58s in. h r8 2.68 2.688s in.

ε r1 6.6 5.2 10sε0 ε r1 6.67 5.14 10.76sε0

h r8 3.8 3.88s in. h r8 3.84 3.848s in.

ε r1 8.4 8.4 12sε0 ε r1 8.38 9.83 12.20sε0

h r8 1.1 1.18s in. h r8 1.24 1.248s in.

ε r1 7.2 7.2 10sε0 ε r1 7.24 8.70 10.35sε0

h r8 1.5 1.58s in. h r8 1.70 1.708s in.

ε r1 6.4 6.4 12sε0 ε r1 6.29 7.30 12.11sε0

h r8 1.1 1.18s in. h r8 1.45 1.458s in.

ε r1 3.1 3.1 10sε0 ε r1 3.17 6.82 12.08sε0

h r8 1.2 1.28s in. h r8 1.79 1.798s in.

ε r1 8.2 8.2 12sε0 ε r1 8.26 7.9 10.87sε0

h r8 4.5 4.58s in. h r8 4.62 4.628s in.

ε r1 7.2 7.2 12sε0 ε r1 7.24 6.89 12.83sε0

h r8 2.2 2.28s in. h r8 1.95 1.958s in.

ε r1 6.5 6.5 15sε0 ε r1 6.62 6.21 15.53sε0

h r8 3 38s in. h r8 2.61 2.618s in.

Note: ε rε0 εa,1 εa,2 εbs and h r8 ha,1 ha,2 8s in.

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determined by the degrees of freedom (number of weights), therefore, one layer can be splitinto several layers whose combination have the same degrees of freedom. However, thereis a trade-off between the available resource and computation time – a multi-layer networkrequires less memory but more time to archive the same goal of one-layer network.

Lastly, the size of neural network is generally based on the available training data sets.If the set is small, according to equation (5.18), the Hm is small as well. However, thenetwork with Hm hidden nodes may not have enough plasticity to perform the requiredrelations inside training data. Because of overfitting, however, having more hidden nodesthan the recommended value Hm is also not desirable. Too much plasticity of network killsthe network generalization ability as well. The redundancy parameter r from 10 to 20 inequation (5.18) is also altered depending on the number of training sets. For example, moreredundancy would be expected when the number of sets is sufficiently large, then r 20 isreasonable to use. If the number of training sets is barely sufficient, then smaller r should beused. Whether or not there is a sufficient number of sets depends on the type of pavementsystem and empirical knowledge.

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Chapter 6

Implementation

In Chapter 5, a backcalculation scheme is proposed and verified with synthetic data generatedbased on the EM model introduced in Chapter 4. The results show that the scheme workswell for interpreting the parameters of pavement system. However, the backcalculationscheme is useful only when it satisfies the field data from GPR survey as well. In fact, indealing with the field data, there are several elements other than the algorithm which playimportant roles. Therefore, in order to guarantee the success of the interpretation of thelayer properties, a standard procedure is provided in this Chapter.

6.1 FRF patterns

It is pointed out that in a constant-parameter linear system, the frequency response function(FRF) is preserved, which means the FRF is purely determined by the system parameters.Given the same parameters, the FRF of the EM model is the same as the FRF of the fielddata, even though the EM model may have slight differences with the actual source fromthe GPR antenna. Because of this important property, the EM model provides a very goodrepresentation of the field data.

Calculation of FRF

In the EM model, the electromagnetic field is generated by a dipole antenna, the smallestcomponent of any GPR antenna. Therefore, the FRF from EM model can be computed byits response Eptq and the presumed dipole source Iptq. In what follows, consider two kindsof GPR surveys, with/without metal plate on the surface, where the survey with metalplate is a calibration procedure for conventional travel-time technique. Therefore, two FRFscharacterizing both surveys can be obtained separately via

FRFmpfq EmpfqIpfq , FRFppfq Eppfq

Ipfq (6.1)

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where Epfq and Ipfq are the Fourier transform of Eptq and Iptq. The subscript “m” and“p” represent the metal plate calibration system and the pavement system, respectively. Itis worth noting that these two different systems have the same input source I.

On the other hand, the FRFs can be calculated from the field data Eptq and the currentsource Iptq of GPR antenna as well, where the FRFs for the metal calibration system andthe pavement system can be computed in a similar fashion via

FRFmpfq EmpfqIpfq , FRFppfq Eppfq

Ipfq (6.2)

where Epfq and Ipfq are the Fourier transforms of Eptq and Iptq. Since the FRF is preserved,given the same parameters, the FRFs in (6.1) and (6.2) are the same.

The reason to compute these two kinds of FRFs is the following. Due to the design of theGPR equipment, not only the desired pulse is contained in the collected GPR record, but alsothe unwanted pulse from the feed point. Figure. 6.1 shows a GPR data scan collected from thepavement, where Tp is the positive peak of the reflection from the surface; Td is the negativepeak of the direct arrival; the unwanted pulse from the feed point is overlapped with thedirect arrival, as shown before the broken line. Therefore, the true direct arrival is impossible

0 50 100 150 200 250 300 350 400 450 500−4000

−2000

0

2000

4000

6000

Sample

Am

plitu

de

GPR data: Pavement

Tp

Td

Figure 6.1: The overlap of the signal from the feed point

to recover from a single scan. However, the ideal full-waveform should contain both the directarrival and the reflection(s) from the layer interface(s), which can be reconstructed from thesynthetic data. It is meaningless to compare two full-waveforms where one has direct arrivaland the other one does not.

As a compromise, one may consider the subtraction of two different waveforms in whichthe direct arrival is removed. Since none of the EM waves can pass through the metal plate,the metal calibration can be interpreted as a pavement survey with no layers underneath.Prompted by the fact that the GPR survey from the metal calibration and the pavement areobtained from an identical equipment setup, the unwanted pulse as well as the ambient noiseshould be largely eliminated upon subtracting from each other. Therefore, the remainder

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represents the difference between a layered pavement and a pavement without layers. Withthe aid of (6.2), one can show that

FRFppfq FRFmpfq Eppfq EmpfqIpfq ∆Epfq

Ipfq . (6.3)

In this way, it provides a valuable comparison between the survey and synthetic data, andgreatly reduces the effect of the noise as well.

6.2 Assumptions

With the aid of Eq. (6.3), it is easy to compute the residual FRFs for synthetic data, butnot for the survey data. Although the ∆Epfq on the right side of Eq. (6.3) can be computedfrom the Fourier transform of the residual waveform from the survey data, the current Ifor the GPR antenna is unknown. Only a coarse estimation of I can be made at this pointby assuming different kinds of pulse functions such as Gaussian pulse, raised cosine andMexican-hat shape. By simulating the waveforms created by those pulses, one can comparethe reflections between the simulation and GPR survey. The data from metal calibration ischosen here because i) the reflection is stronger for better comparison, and ii) it is a knowncondition which can be simulated by the EM model. Then, the reflection pulse which isclosely matched in terms of the amplitude and shape indicates that its source pulse is themost similar one to the real current source.

As a result, the matched source pulse is the Mexican-hat shape pulse as shown in Fig. 6.2,and it will be applied in the following procedures. It should be noted that there is noguarantee about the presumed source which may lower the accuracy of the interpretation.

Besides the assumption for the current source, it is found that the height of the effectivetime-zero of the antenna cannot be determined by the given spacing information of the GPRbox. In the manual of the model 4108 antenna, the calculation for the reflection from surfaceis given as pTp Td TcqV 2

?h2 x2 (6.4)

where Tp and Td are shown in Fig. 6.1 and

• Tc = time constant that corrects for the difference in the arrival time of the positivepeak of the direct-coupling and the true time-zero (This value was back-calculatedusing the same equation and a series of measurements over a metal plate obtained atknown heights);

• Tp = time corresponding to positive peak of pavement surface reflection arrival;

• Td = time corresponding to negative peak of direct-coupling arrival;

• V = propagation velocity of radar waves in free space;

• h = antenna height above pavement or metal plate;

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5 5.5 6 6.5 7

x 10−9

−4000

−2000

0

2000

4000

6000

Time

Am

plitu

de

The reflections from metal plate

SimulationGPR survey

Figure 6.2: Comparison of reflection between simulation and GPR survey

• x = 1/2 bistatic separation between transmitting and receiving antennas (for model4108, 2x 13.189 in.).

The question comes up at this point is whether the Tc can be determined by the givenparameters, i.e., h and x, or vice versa. Unfortunately, the value of Tc is also unrevealed bythe manufacturer. From the setup, the GPR box is generally lifted up at 18 20 in. abovethe ground. In addition, the vertical spacing inside the GPR box is 15.5 in., resulting in ah 33.5 35.5 in. A simple verification can be done based on the GPR survey as shownin Fig. 6.3 where Tp 8.1 ns and Td 5.36 ns. If one assumes that the Tc is determined by

0 2 4 6 8 10 12 14 16 18 20−4000

−2000

0

2000

4000

6000

Time (ns)

Am

plitu

de

GPR data: Metal plate

Td

Tp

T0

Figure 6.3: GPR survey of metal plate

h and x, when h 33.5 in. and 2x 13.189 in., the Tc ends up to be 3.04 ns, which means

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the effective time-zero is at t 2.32 ns (T0 in Fig. 6.3). It seems not possible since T0 iseven before the unwanted pulse from the feed point.

On the other hand, one can assume that the constant Tc determines h and x. Withthe aid of Tc, it permits the real-time calculation of the GPR antenna height above theground during the survey. However, it requires a series of experiments based on Eq. (6.4) todetermine the constant Tc. In addition, because the type of the current source also affectsthe value of Tc, the difficulty to determine Tc is increased due to the uncertainty of thesource. To overcome this issue, one may skip Tc to directly determine the height of effectivetime-zero by comparing the data between the simulation and survey as before.

6.3 Implementation procedures

As stated earlier, the FRF pattern characterizing a linear system is determined by the systemparameters. A back-analysis scheme based on artificial neural network (ANN) is employedto interpret the parameters of pavement properties from the FRF patterns. By means of(6.3), the neural network is trained with the residual FRF calculated from the EM model.Afterward, the ∆EpfqIpfq calculated from the survey data is interpreted by the trainednetwork to produce the pavement properties, by using the asphalt layer dielectric constantεA, the asphalt layer thickness hA and the base layer dielectric constant εB.

Collection of field data

The SIR20 is the unit operated by Mn/DOT to collect GPR data. The settings for theparameters are shown in Fig. 6.4(a). As shown, the recommended configuration type forhighway scan is “HIGHWAY”. Upon choosing this scan type, the pre-set values of sam-ples/scan = 512, scan/sec = 153. The other values can be input depending on the situationsuch as the DieConstant, scan/ft, and ft/mark. It should be denoted that the default rangevalue of the window is 20 ns for highway configuration. The values of samples/scan and rangedetermine the time interval between the data points. For example, a 512 samples/scan witha range of 20 ns, the samples are collected at rate of 20ns511 0.03911 ns/sample.

Before collecting data, the provided IIR filters have to be turned off in order to obtain thetrue amplitude of the radar response. In addition, there has to be only one gain point (seeFig. 6.4(b)), i.e., uniform gain throughout the scan. The uniformity of gain value is importantin terms of comparing the amplitude with different sections of the scans. Otherwise, it ismeaningless to compare normal waveform with distorted one.

Because the height of the source is a fixed parameter during the backcalculation, theeffective height of the GPR antenna has to equal the assumed height for simulation. It isnot possible to calculate the effective height in real-time at the time being. The only way tolocate the effective height is to manually match the simulation with the metal calibration.Due to the fluctuation when moving, however, the pavement survey is valid only when theGPR antenna is lifted up to the same elevation of the calibration height.

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(a) Config Type: Highway, Samples/Scan=512, Auto Gain Level=0

(b) Number of Points=1, Auto Gain Servo(on)

Figure 6.4: Two steps to setup project

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To facilitate the implementation of this method, a graphic user interface is designed forconvenient operation, called “GopherGPR” as shown in Fig. 6.5.

With the aid of this panel, it allows quickly process of the GPR data from the roadsurvey. The manual for the software is provided in the Appendix C.

6.4 Summary

Based on the limitations of the existing system, there are some modifications of the imple-mentation procedure proposed in this chapter compared to the backcalculation scheme inChapter 5.

Since the direct arrival is immersed into the response from the feed point, it is impossibleto recover the ideal scan from the regular GPR survey. Therefore, the idea to subtract themetal calibration from the regular GPR survey is accommodated here, which provides avaluable comparison between the survey data and simulation and reduces the signal/noiseratio as well.

Since some inside information about the existing system is unavailable, two major as-sumptions have to be pointed out during the proposed implementation procedure that 1)the Mexican-hat pulse is applied as the excitation source for GPR antenna, and 2) the ef-fective height of the GPR survey is obtained via manually comparison between the metalcalibrations from the survey and simulation. Due to the limited time of this project, it isnot possible to verify these two assumptions with all situations.

Last but not least, given new valid effective height, the network has to be re-trained sincethe height is a fixed parameter in training the neural network.

In terms of the interpreted parameters, the accuracy of all the values are dependentupon the accuracy of the effective height. In addition, the value of the base layer dielectricconstant would be less reliable than the other two, i.e., the thickness and dielectric constantof the asphalt layer. The reason may be explained by the following:

• The top layer dielectric constant is subjected to the correct value of the height ofthe source. One may imagine two surface reflections with same amplitude. The onewith smaller antenna height means less dissipation of energy, resulting in a smallerinterpreted value of the top layer dielectric constant. Vise versa, the larger heightwould be interpreted as a larger value of the dielectric constant in the top layer.

• It is obvious that the thickness measurement depends on the layer velocity and timemark in the scan at the top and bottom of the layer, where the layer velocity is afunction of the layer dielectric constant. Due to the previous reason, the dielectricconstant is affected by the accuracy of the source height, as well as the layer velocity,the layer thickness is subjected to the correct value of the height.

• For the base layer, the situation is worse because the errors from the misinterpretationof the parameters in the layer above will be added to the total error in this layer.

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Figure 6.5: GopherGPR panel

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However, all these can be improved as long as the effective height can be obtained asaccurate as possible.

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Chapter 7

Summary

The GPR methodology is similar to shallow seismic reflection surveys in the sense that bothtechniques use energy reflected from (subsurface) material interfaces. Employing this anal-ogy, the existing GPR data analysis can be enhanced by drawing from the extensive bodyof literature and experience in shallow seismic reflection. This research aims to generalizeand improve upon the related development. In particular, a new algorithm based on elec-tromagnetic wave propagation in layered (as opposed to single, AC-layer) pavement systemhas been developed, which could be used as a realistic basis for the interpretation of GPRmeasurement taken on a pavement profile.

Drawing from the invaluable solutions of the EM model in layered system, more accurateinterpretation of GPR images in terms of dielectric constant and thickness in different layerscan be obtained via the backcalculation scheme based on artificial neural network (ANN)and frequency response function (FRF). The proposed backcalculation scheme is verifiedwith synthetic data where the results show that the scheme works well in interpreting theparameters of pavement system. However, it should be noted that different layer structuresrequire different setup for the scheme. Otherwise, it is out of the capability of what thebackcalculation can do.

In addition, it is found that the accuracy of the interpreted pavement parameters aregreatly affected by the effective height of the source, which can only be determined bymanually comparing the survey scan and simulation at the time being. In this project, theeffective height of the illustration sample has been determined and only applicable to thesample metal calibration. Given a new metal calibration, the effective height needs to bemanually determined again.

In the end, the research results would help districts and local governments to performcontinuous evaluation of the pavement thickness and density quickly and more accurately.In turn, this benefit will inherently increase construction efficiency and the reduce cost ofpavement construction and maintenance.

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Appendix AData Set of Field Sample

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Table A-1: Data from the MnDOT

Lab ID Air Dry Dry SSD Imm. VWC Gmb Gmm %Gmm ε

1503 1019.1 1017.6 1020.1 598 0.25 2.411 2.51 96.0 4.91504 1090.1 1017.6 1020.1 598 0.25 2.411 2.51 96.0 4.91505 956.2 954.9 957.4 553 0.26 2.361 2.512 94.0 51506 779.2 777.7 780.4 446.4 0.35 2.328 2.52 92.4 4.81507 979.4 976.5 980.2 572.6 0.38 2.396 2.511 95.4 5.21508 977.3 976 978 563.4 0.20 2.354 2.457 95.8 4.71509 847.8 841.2 848.4 481.7 0.86 2.294 2.494 92.0 5.31510 947.2 944.6 947.8 540.6 0.34 2.320 2.494 93.0 5.21511 885.1 884.1 886.1 507.6 0.23 2.336 2.494 93.7 5.31512 1752 1743.2 1754 1014.7 0.62 2.358 2.497 94.4 5.51513 903.7 896.2 905.4 510 1.03 2.267 2.494 90.9 4.81523 1608.5 1607 1609.6 935.1 0.16 2.383 2.468 96.5 4.41524 1002.9 1002.1 1003.6 584.5 0.15 2.391 2.482 96.3 5.31525 574.2 573.4 574.8 332 0.24 2.362 2.482 95.1 4.91526 890.1 887.5 890.9 511.1 0.38 2.337 2.496 93.6 4.31527 678.8 676.8 679.1 392.3 0.34 2.360 2.496 94.5 3.51528 646.2 645.1 647.2 374.2 0.33 2.363 2.496 94.7 4.41529 703.9 701 705 403.7 0.57 2.327 2.496 93.2 4.21530 1141.9 1139.7 1142.6 657.3 0.25 2.348 2.477 94.8 4.21531 1164.9 1162.2 1166.6 672.8 0.38 2.354 2.477 95.0 51532 1196.4 1195 1197.7 698.4 0.23 2.393 2.477 96.6 4.21533 1097.9 1096.2 1098.6 638.6 0.22 2.383 2.477 96.2 4.91534 1116.3 1114.7 1117.8 648.7 0.28 2.376 2.477 95.9 5.21535 1133.9 1131.3 1135.7 646.6 0.39 2.313 2.488 93.0 4.91536 635.6 629.3 636.4 356.6 1.13 2.249 2.518 89.3 4.61537 941.1 933 942.6 543.8 1.03 2.340 5.41538 993 926.1 935 533 0.96 2.304 5.11539 1073.6 1064 1075.6 613.1 1.09 2.301 4.91540 739.3 736.9 739.9 409.7 0.41 2.232 4.71541 906 901 907.3 507.5 0.70 2.254 4.81542 743.1 738.6 746.6 403.4 1.08 2.152 4.71543 658.1 654.9 660.3 362.6 0.82 2.200 4.31544 1475.5 1473.6 1476.6 857.8 0.20 2.381 2.472 96.3 5.21545 1150.1 1149 1151.2 668.5 0.19 2.380 2.472 96.3 5.2

A-1

Page 78: Pavement Evaluation Using Ground Penetrating … Evaluation Using Ground Penetrating Radar Technical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. MN/RC 2008-10

Table. A-1 Data from MnDOT (cont.)Lab ID Air Dry Dry SSD Imm. VWC Gmb Gmm %Gmm ε

1546 1123.1 1122.2 1123.7 657.3 0.13 2.406 2.472 97.3 51547 766.2 762.4 767.6 437.6 0.68 2.310 2.512 92.0 51548 744.5 741.4 746.2 425 0.65 2.308 2.512 91.9 5.11549 825.9 825.7 826.8 481.4 0.13 2.391 2.498 95.7 4.21550 858 854.3 859.1 485.7 0.56 2.288 2.498 91.6 4.61551 854 850.9 856 489.5 0.60 2.322 2.498 92.9 4.71552 882.3 876.2 887.4 503.6 1.28 2.283 2.498 91.4 4.61553 1212.3 1201.7 1214.1 684 1.03 2.267 2.267 100.0 5.61554 1412.7 1400.5 1415 799.1 1.04 2.274 5.31557 970.2 969.4 970.9 573.1 0.15 2.437 2.558 95.3 5.31558 1213.2 1208.8 1214.5 711.2 0.47 2.402 2.558 93.9 5.91559 1013.2 1007 1014.7 586.4 0.76 2.351 2.558 91.9 6.21560 1115 1102.8 1116.7 638.2 1.26 2.305 2.558 90.1 6.21561 722.5 714.5 724.4 408.1 1.39 2.259 2.563 88.1 6.41562 1183.7 973.3 986.7 561 1.38 2.286 2.563 89.2 5.71563 976.5 972.4 978.1 565.6 0.59 2.357 2.563 92.0 6.11564 989.2 978.8 992.3 566.3 1.38 2.298 2.563 89.6 6.51565 867.8 866.7 869.3 492 0.30 2.297 2.522 91.1 5.051566 761.7 760.9 763 437.7 0.28 2.339 2.522 92.7 5.41567 854.3 852.7 855.9 484.2 0.38 2.294 2.522 91.0 5.091568 1021.5 1020.9 1022.4 596.6 0.15 2.398 2.492 96.2 5.031569 998.1 997.8 999.4 582.1 0.16 2.391 2.488 96.1 5.341570 1847.8 1840.8 1849.4 1056.1 0.47 2.320 2.494 93.0 51571 1598.1 1586.5 1603.9 893 1.10 2.232 2.498 89.3 4.51572 692.9 690.4 694.1 396.5 0.54 2.320 2.535 91.5 5.221573 934.1 928.2 937 531.3 0.95 2.288 2.535 90.3 5.61574 631.7 627.6 634.2 358.5 1.05 2.276 2.537 89.7 5.271575 687.7 681.2 692.5 381.9 1.66 2.193 2.537 86.4 4.61576 885.6 879.9 886.8 502 0.78 2.287 2.514 91.0 5.221577 793.5 790.4 794.9 452.5 0.57 2.308 2.514 91.8 5.481578 608.1 605.2 609.5 346.8 0.71 2.304 2.514 91.6 5.481579 915.5 914.1 918.3 523.9 0.46 2.318 2.53 91.6 5.211580 766.7 765.2 769.2 437.2 0.52 2.305 2.53 91.1 4.251581 770.9 769.7 772.9 442.2 0.42 2.327 2.53 92.0 5.411582 986.2 978.4 987.4 565.1 0.92 2.317 2.508 92.4 4.681583 1081.9 1069.3 1090.6 607.2 1.99 2.212 2.508 88.2 5.281584 932.8 931.4 933.7 533 0.25 2.324 5.21585 1042.6 1040.2 1044.4 592.1 0.40 2.300 2.504 91.8 5.231586 847.2 845.3 848.3 484.2 0.35 2.322 2.504 92.7 5.451587 803.9 801.7 805 460.6 0.41 2.328 2.504 93.0 6.241588 952.8 951.4 953.8 550.7 0.25 2.360 2.504 94.3 5.531589 822.8 821.9 823.9 477.7 0.24 2.374 2.504 94.8 5.181590 935.4 931.8 936.3 533.2 0.48 2.312 2.516 91.9 6.091591 879.6 877.6 880.8 514.9 0.36 2.398 2.516 95.3 5.61592 1119.2 1117.5 1120.1 652.1 0.23 2.388 2.516 94.9 5.361593 895.1 893.4 896.6 516.5 0.36 2.350 2.516 93.4 5.611625 1124.4 1120.9 1125.3 653.5 0.39 2.376 2.488 95.5 5.411626 1148.7 1144.2 1150.4 659 0.54 2.328 2.512 92.7 4.91627 722.8 721.8 724.4 396.4 0.36 2.201 2.506 87.8 3.71628 930.6 929.3 932 521.7 0.29 2.265 2.506 90.4 4.891629 821.2 820.4 823.4 469.4 0.37 2.318 2.506 92.5 4.831630 1205.7 1204.9 1207.5 690.6 0.22 2.331 2.501 93.2 5.511632 1001.1 1000 1003.7 561.4 0.37 2.261 2.503 90.3 5.131633 1061.5 1059 1065.4 599.2 0.60 2.272 2.503 90.8 5.571634 734.9 733.7 736.9 415.4 0.44 2.282 2.498 91.4 5.221635 606.8 605.9 608 353.1 0.35 2.377 2.482 95.8 61636 731.1 730.1 731.9 419.1 0.25 2.334 2.482 94.0 5.541637 734.7 732.8 736 419.9 0.44 2.318 2.482 93.4 5.73

A-2

Page 79: Pavement Evaluation Using Ground Penetrating … Evaluation Using Ground Penetrating Radar Technical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. MN/RC 2008-10

Table. A-1 Data from MnDOT (cont.)Lab ID Air Dry Dry SSD Imm. VWC Gmb Gmm %Gmm ε

1638 743.3 736.5 745 421.7 1.15 2.278 2.542 89.6 5.621639 1180.7 1178.4 1182.1 688.4 0.31 2.387 2.525 94.5 5.421640 1403.2 1399.2 1403.9 816.6 0.34 2.382 2.525 94.4 5.411641 1270.4 1267.8 1271.7 756.4 0.31 2.460 2.525 97.4 5.51642 1004.6 1001.3 1006 586 0.47 2.384 2.525 94.4 5.321643 1579.2 1575.4 1580.4 926.4 0.32 2.409 2.484 97.0 5.641644 814.4 813.8 815.2 475 0.17 2.392 2.481 96.4 5.781645 676.4 672.8 677.3 382.3 0.67 2.281 2.481 91.9 4.921646 1425.6 1424.2 1427.9 834.1 0.26 2.398 2.516 95.3 5.861647 1437.1 1431.1 1439 822.6 0.55 2.322 2.516 92.3 5.91648 1517.4 1515.6 1518.7 887.1 0.20 2.400 2.516 95.4 6.11649 979.8 977.2 980.9 570.1 0.38 2.379 2.516 94.5 5.261658 1209.5 1205.9 1210 699.1 0.34 2.360 4.61659 1532.2 1531.3 1533 888.8 0.11 2.377 5.351660 1238.4 1235.3 1239.1 711.7 0.31 2.342 5.61661 1244.4 1240.8 1245.6 710.5 0.39 2.319 5.81668 895.5 887.7 898 506.2 1.16 2.266 5.671669 751.7 743.2 753 428 1.32 2.287 4.621670 722.4 715.4 725.1 408.8 1.36 2.262 5.431671 553.7 545.5 556.8 309.8 2.07 2.209 4.141672 790 778.4 793.8 439.4 1.98 2.196 4.681673 878.9 874.7 880 504.1 0.61 2.327 5.061674 246.7 244.8 253.2 139.4 3.43 2.151 3.711675 979.6 978 981 557.2 0.31 2.308 2.478 93.1 5.741676 826.6 825.5 827.8 482.5 0.28 2.391 2.478 96.5 5.481677 947.8 945.2 948.5 545.6 0.35 2.346 2.478 94.7 5.591678 1175.3 1174 1176.4 680.4 0.20 2.367 2.492 95.0 5.51679 842.7 841.2 843.7 481.4 0.30 2.322 2.492 93.2 5.081680 1048.8 1046.9 1049.4 607.2 0.24 2.367 2.492 95.0 5.071681 979.4 978.1 980.2 570.9 0.21 2.390 2.492 95.9 5.391682 1232.7 1225.5 1233.6 705.7 0.66 2.321 5.321683 1017.1 1012.8 1017.6 590.1 0.47 2.369 4.531684 908.8 905.2 910.3 513.7 0.56 2.282 5.981685 966.6 965.2 967.5 573.9 0.24 2.452 5.221701 1060.2 1058.7 1061.1 614.8 0.23 2.372 2.496 95.0 5.211702 1275.2 1272.4 1276.2 746.6 0.30 2.403 2.498 96.2 5.61703 935.7 931.4 936.6 542.5 0.56 2.363 2.482 95.2 5.731704 687.7 689.7 688.5 392.7 -0.17 2.332 2.482 93.9 4.921705 1213.9 1213.2 1215 709.1 0.15 2.398 2.496 96.1 5.41706 1011.4 1010.7 1011.8 594.8 0.11 2.424 2.496 97.1 51707 1818.8 1817.7 1819.9 1072 0.12 2.430 2.496 97.4 5.81708 810.5 809.9 811.1 470.8 0.15 2.380 2.471 96.3 4.921709 1127.5 1126.8 1127.9 659.7 0.10 2.407 2.471 97.4 5.021710 843.7 843.2 844.7 483.6 0.18 2.335 2.492 93.7 4.991711 950 949.6 952.1 542.2 0.26 2.317 2.496 92.8 4.511712 1600.5 1595.4 1601.7 922.8 0.39 2.350 2.501 94.0 4.341713 1271 1270.7 1272.4 726 0.13 2.326 2.474 94.0 5.241714 887 886.6 889.1 509.1 0.28 2.333 2.471 94.4 5.11

A-3

Page 80: Pavement Evaluation Using Ground Penetrating … Evaluation Using Ground Penetrating Radar Technical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. MN/RC 2008-10

Table A-2: Data from lab asphalt sample

Sample Weight T Permittivity GVCID [g] [h] 1st [J] 2nd [J] 3rd [J] Mean [J] Sqrt [J] [%]

7219.2 Dry 4.3 4.6 4.1 4.3 2.1 -#3 58-28B 7231.5 0 4.6 4.6 4.5 4.6 2.1 0.17

4%AV Opt L 7230.8 1 4.6 4.6 4.5 4.6 2.1 0.167230.4 2 4.5 4.6 4.4 4.5 2.1 0.167230.6 3 4.4 4.5 4.4 4.4 2.1 0.16

7236.9 Dry 4.6 4.6 4.5 4.6 2.1 -#12 58-28B 7243.1 0 4.8 4.7 4.7 4.7 2.2 0.094%AV Opt L 7242.3 1 4.7 4.5 4.6 4.6 2.1 0.07

7242.6 2 4.6 4.5 4.5 4.5 2.1 0.087242.0 3 4.6 4.5 4.6 4.6 2.1 0.07

7257.2 Dry 3.9 3.6 3.8 3.8 1.9 -#9 58-28A 7276.6 0 4.0 4.1 4.2 4.1 2.0 0.27

4%AV Opt L 7260.7 1 4.0 4.0 3.9 4.0 2.0 0.057260.6 2 3.9 3.8 3.8 3.8 2.0 0.057260.1 3 4.0 3.9 3.8 3.9 2.0 0.04

7239.6 Dry 4.3 4.2 4.3 4.3 2.1 -#4 58-28A 7287.1 0 4.3 4.5 4.5 4.4 2.1 0.66

7%AV Opt L 7284.0 1 4.4 4.4 4.5 4.4 2.1 0.617283.5 2 4.3 4.2 4.4 4.3 2.1 0.617282.2 3 4.3 4.4 4.4 4.4 2.1 0.59

7164.4 Dry 3.9 3.6 3.8 3.8 1.9 -#9 58-28A 7172.0 0 4.0 4.1 4.2 4.1 2.0 0.11

4%AV Opt G 7169.8 1 4.0 4.0 3.9 4.0 2.0 0.087169.8 2 3.9 3.8 3.8 3.8 2.0 0.087169.3 3 4.0 3.9 3.8 3.9 2.0 0.07

7220.7 Dry 3.8 3.8 3.9 3.8 2.0 -#10 58-28A 7229.4 0 4.1 4.2 4.3 4.2 2.0 0.127%AV Opt G 7227.2 1 4.0 4.2 4.3 4.2 2.0 0.09

7226.8 2 3.8 3.9 3.8 3.8 2.0 0.087226.6 3 3.8 3.9 3.8 3.8 2.0 0.08

GVC = Gravimatric Water ContentAV = Air voids

Opt = optimum propotion propotion of asphalt binderL/G = limestone/granite

A-4

Page 81: Pavement Evaluation Using Ground Penetrating … Evaluation Using Ground Penetrating Radar Technical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. MN/RC 2008-10

Tab

leA

-3:

Dat

afr

omfine-

grai

ned

soil

:pri

smat

icsa

mple

Conta

iner

Perm

itti

vity

Conducti

vity

Wate

rC

onte

nt

IDH

igh

[J]

Low

[J]

Media

n[J

]M

ean

[J]

Sqrt

[?J]

Cov

[%]

Hig

h[S

/cm

]Low

[S/cm

]M

edia

n[S

/cm

]M

ean

[S/cm

]C

ov

[%]

[%]

A15

100

27.2

20.8

24.1

24.0

4.9

13.3

90

44.0

93.5

15.6

B24

103

47.2

34.9

40.0

40.8

6.4

14.6

44

823

24.4

54.8

23.6

B19.5

103

42.5

18.7

33.0

31.7

5.6

24.1

108

014

21.3

144.4

19.2

B16

103

30.2

11.2

28.7

26.3

5.1

22.9

44

018

20.1

93

17.4

B27

98

45.4

22.9

32.3

32.2

5.7

23.4

21

03

5.6

126.3

26.1

B22

98

49.4

31.3

40.3

40.3

6.3

14.1

135

052

54.4

79.4

22.0

B16

98

24.0

16.0

20.9

20.1

4.5

14.3

48

18

23

26.6

42.6

16.3

C12

103

19.3

11.2

17.6

16.4

4.0

16.2

6.4

60

828.0

25.9

11.3

C9.5

103

17.9

14.4

16.4

16.2

4.0

6.4

60

828

25.9

67.6

9.4

C8

103

17.7

10.9

16.2

15.6

3.9

14.1

46

013

16.3

96.6

8.4

C13

98

18.6

10.5

17.2

15.8

4.0

20.6

43

00

8.6

223.6

12.4

C10

98

15.3

13.9

14.7

14.5

3.8

3.7

90

261

45.9

69.8

10.1

C8

98

18.0

15.8

17.1

17.0

4.1

3.8

74

25

49

52.9

28.6

8.4

D16

103

45.6

35.4

38.2

39.6

6.3

9.3

114

357

61.6

59.9

16.3

D13

103

38.6

30.0

34.4

34.6

5.9

9.4

54

221

22.7

69.6

13.3

D11

103

31.8

23.0

26.1

26.9

5.2

13.1

26

07

7.6

108.4

10.6

D18

98

49.4

28.5

43.4

41.8

6.5

16.7

211

316

60.1

142.9

18.6

D14

98

36.6

21.2

30.0

30.1

5.5

18.7

185

39

96

105.5

44.5

14.2

D11

98

28.7

17.0

25.5

24.4

4.9

15.5

61

028

23.1

92.3

10.7

Tab

leA

-4:

Dat

afr

omfine-

grai

ned

soil

:cy

linder

sam

ple

Conta

iner

Perm

itti

vity

Conducti

vity

Wate

rC

onte

nt

IDH

igh

[J]

Low

[J]

Media

n[J

]M

ean

[J]

Sqrt

[?J]

Cov

[%]

Hig

h[S

/cm

]Low

[S/cm

]M

edia

n[S

/cm

]M

ean

[S/cm

]C

ov

[%]

[%]

B19.5

103

38.9

37.5

38.3

38.2

6.2

1.8

18

417

13.0

60.1

18.5

B16

103

39.0

37.9

38.3

38.4

6.2

1.4

13

45

7.3

67.3

16.5

B27

98

42.5

39.7

42.0

41.4

6.4

3.6

90

56

81

75.7

23.3

23.1

B22

98

38.0

37.0

37.2

37.4

6.1

1.4

90

56

81

75.7

23.3

21.2

Spec

#1

C8

98

16.7

15.8

16.3

16.3

4.0

2.8

24

12

23

19.7

33.9

8.4

D16

103

34.8

24.1

34.1

31.0

5.6

19.3

26

10

21

19.0

43.1

15.6

D13

103

36.1

31.5

35.4

34.6

5.9

6.0

60

54.0

67.7

13.1

D11

103

30.9

29.7

30.1

30.2

5.5

2.0

10

10.7

86.6

10.5

D18

98

38.4

33.1

34.7

35.4

5.9

7.7

81

18

57

52.0

61.1

18.3

D14

98

36.4

36.0

36.1

36.2

6.0

0.6

40

11

30

27.0

54.6

14.3

D11

98

25.8

20.6

21.3

22.6

4.8

12.5

10

36

6.3

55.5

10.7

B19.5

103

35.4

32.6

33.9

34.0

5.8

3.4

27

09

4.5

139.1

18.3

B16

103

36.6

35.2

36.2

36.0

6.0

2.0

20

18

19

19.0

5.3

16.4

B27

98

38.1

36.4

37.3

37.3

6.1

3.2

106

80

93

93.0

19.8

24.0

B22

98

44.0

43.4

43.9

43.8

6.6

0.7

46

36

39

36.0

14.7

21.5

Spec

#2

C8

98

19.1

18.8

19.0

19.0

4.4

0.8

18

10

14

15.0

28.2

8.2

D16

103

31.4

30.1

31.0

30.8

5.5

2.2

10

78

8.0

18.3

15.6

D13

103

36.1

32.3

34.8

34.4

5.9

4.5

11

04

1.0

127.6

13.2

D11

103

30.9

30.2

30.7

30.6

5.5

1.2

31

21.0

69.3

10.6

D18

98

35.6

34.4

34.4

34.8

5.9

2.0

21

13

18

21.0

25.2

18.4

D14

98

33.3

31.7

32.4

32.5

5.7

2.5

21

13

18

21.0

25.2

13.6

D11

98

25.3

24.4

25.0

24.9

5.0

1.8

50

33.0

94.4

10.7

A-5

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Appendix BElectromagnetic Analysis in Layered

System

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To solve the electromagnetic field in layered structure, the Maxwell’s equations for linearmaterial are rewritten here

∇ E %

ε∇ H 0

∇ E µBHB t

∇H J εBEB t

(B-1)

where µ µ0 is usually adopt for most non-magnetic materials. By means of time-harmonicrepresentation, Epr, tq Eprqeiωt and Hpr, tq Hprqeiωt, where ω circular frequency.Hence, the Maxwell’s equations are replaced by

∇ E %

ε∇ H 0

∇ E iωµH ∇H J iωεE

(B-2)

where the time factor eiωt is suppressed. To uncouple the E and H fields, upon taking thecurl of the third equation in (B-2), and substituting with the fourth equation, one obtainsthat

∇∇ E iωµpJ iωεEq (B-3)

By means of the identity ∇∇ E ∇p∇ Eq ∇2E and the equation of continuity

∇ J B%B t 0, (B-4)

Eq. (B-3) is rearranged as Helmholtz equation

p∇2 k2qE iωµ

¯I ∇∇

k2

J (B-5)

where k2 ω2µε is the wave number and ¯I is unit vector. In what follows, consider a Green’sfunction satisfies p∇2 k2qgpr, r1q δpr r1q (B-6)

where the operator ∇ only works on r. It can be shown that the solution of E takes theform via the Green’s function like that

Eprq iωµ

»V 1

gpr, r1q

¯I ∇1∇1k2

Jpr1qdr1. (B-7)

It can be verified upon substituting (B-6) and (B-7) into (B-5) where operator ∇ can be put

B-1

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into the integral like that

iωµ

»V 1

δpr r1q

¯I ∇1∇1k2

Jpr1qdr1 iωµ

¯I ∇∇

k2

Jprq. (B-8)

In the following, by means of the identity ∇pgfq f∇g g∇f and ∇ pgFq ∇g Fg∇ F, one may show that

g∇1∇1 Jpr1q ∇1

g∇ Jpr1q∇1g∇ Jpr1q

∇1

g∇ Jpr1q∇1

∇1g Jpr1q

Jpr1q ∇1∇1g.

(B-9)

In addition, assume that there is neither charge inside the volume (% 0 in V 1) nor surfacecurrent (J 0 on BV 1), it is followed by that the volume integrals of the first two terms onthe right in (B-9) are vanished.»

V 1∇1

g∇ Jpr1q

dr1

»V 1∇1

gB%B t

dr1 0

»V 1∇1

∇1g Jpr1q

dr1

»BV 1

∇1g Jpr1q dΩ1 0

(B-10)

As a result, with the aid of (B-9) and (B-10), Eq. (B-7) becomes to that

Eprq iωµ

»V 1

Jpr1q

¯I ∇1∇1k2

gpr, r1qdr1 (B-11)

Denote ssG as the dyadic Green’s function,

ssG

gxx 0 0

0 gyy 0

gxz gyz gzz

(B-12)

where, for instance, gxz represents the response in z-direction due to the x-direction excita-tion. The reason for the missing component of gxy is due to the natural restriction from theinterface. By substituting (B-12) back into (B-11) and denoting that

Jprq I∆lδpr r1q (B-13)

B-2

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the electric field due to infinitesimal current source is readily obtained in terms of the dyadicGreen’s function via

E iωµI∆l

k2

k2 ∇∇ ssG. (B-14)

In this project, due to the difficulty in obtaining the information about the antennasource, the transmitter is presumed as a vertical line antenna, as well as the receiver. Asa result, only the vertical source is considered, and the received electric field is in verticaldirection only as well.

Vertical source

For a vertical source, viz, gprqez gzzprqez 0, the electric field is determined by that,

Eprq iωµI∆l

k2

k2 ∇∇ gprqez. (B-15)

After some manipulation, the components of the electric field in (B-15) can be shown incylindrical coordinate as

Eρpρ, zq iωµI∆l

k2

BB ρ

BB z

gpρ, zq

Ezpρ, zq iωµI∆l

k2

k2 B2

B z2

gpρ, zq

(B-16)

where r2 ρ2 z2 and the Eθ does not exist due to symmetry.Introduce the nth-order Hankel transform which is defined as a pair of integral transforms

via

fnpξ, zq 8»0

fpρ, zqJnpξρqρdρ

fnpρ, zq 8»0

fpξ, zqJnpξρqξdξ

(B-17)

where Jn is the nth-order Bessel function of the first kind. Upon applying the 1st-order and0th-order Hankel transforms to Eρ and Ez, respectively, the transformed field of (B-16) can

B-3

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be shown as

E1ρpξ, zq iωµI∆l

k2

BB z

8»0

BB ρ

gpρ, zq ρJ1pξρqdρ

E0z pξ, zq iωµI∆l

k2

k2 B2

B z2

8»0

gpρ, zq ρJ1pξρqdρ

(B-18)

where the superscript indicates the order of Hankel transform. By means of the identitiesthat B

B ρJ0pξρq ξJ1pξρq, J1pξρq ρ

BB ρ

J1pξρq ρξJ0pξρq (B-19)

it can be observed that

8»0

BB ρ

gpρ, zq ρJ1pξρqdρ ξ

8»0

gpρ, zq ρJ0pξρqdρ ξ g0pξ, zq. (B-20)

In addition, the partial derivative with respect to z in the above is in fact a full derivative,i.e., B

B zg0pξ, zq d

dzg0pξ, zq g1pξ, zq (B-21)

where the superscript for the order of Hankel transform is suppressed.In what follows, upon applying the 0th-order Hankel transform to Eq. (B-6), it is readily

to show the following d2

dz2 k2 ξ2

g 0 (B-22)

in which the solution of g is obtained via

gpξ, zq Apξqeγz Bpξqeγz (B-23)

where γ2 ξ2 k2 and the depth-independent coefficients Apξq and Bpξq are determined bythe boundary conditions. Therefore, with the aid of (B-20) and (B-21), Eq. (B-18) becomesto that

Eρpξ, zq iωµI∆l

k2ξ g1pξ, zq

Ezpξ, zq iωµI∆l

k2ξ2 gpξ, zq

(B-24)

where the superscript for the order of Hankel transform is suppressed. Note that in (B-24)the g is obtained via the 0th-order Hankel transform, as well as the Ez. The Eρ, however, isobtained via the 1st-order Hankel transform.

B-4

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Layered solution

Consider a n parallel layered system in between two half-spaces with a cylindrical coordinatesystem set at the bottom of the upper half-space (see Fig. B-1). All the layers are labeledfrom 0 (upper half space, air) to n 1 (lower half space). The jth layer denoted as domainLj pj 0, 1, , n 1q is characterized by its material parameters: dielectric constant εj,permeability µj, conductivity σj and layer thickness hj zj zj1 where zj and zj1 are thedepths of its upper and lower interfaces.

Figure B-1: Schematic of layered structure

When crossing the layer interface, the electric field can be described as that the compo-nent of E parallel to the interface is continuous, whereas the component of E perpendicularto the interface has a jump. Since the layers are horizontally placed in the setup, one can

B-5

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obtain the following on the jth interface (z zj)

Eρ,jpρ, zjq Eρ,j1pρ, zjq, εjEz,jpρ, zjq εj1Ez,j1pρ, zjq. (B-25)

By means of Hankel transform in (B-17), Eq. (B-25) is transformed into that

Eρ,jpξ, zjq Eρ,j1pξ, zjq, εjEz,jpξ, zjq εj1Ez,j1pξ, zjq (B-26)

or in terms of gpξ, zq via (B-24)

1

εj

g1jpξ, zjq 1

εj1

g1j1pξ, zjqgjpξ, zjq gj1pξ, zjq

(B-27)

With the aid of (B-23) and (B-24), the solutions for the layers without excitation sourceare readily obtained except those depth-independent coefficients Ajpξq and Bjpξq. In thelth layer where zl1 s zl, it is convenient to decompose the solution into portions withand without the source. As a result, in domain Ll, one may express the solution throughdepth-dependent coefficients in terms of the depth-independent coefficients Apξq and Bpξq

Alpξ, zq Alpξq Hps zqAl,ppξqBlpξ, zq Blpξq Hpz sqBl,ppξq

(B-28)

where H denotes the Heaviside step function; Al,ppξq and Bl,ppξq are coefficients for particularsolutions due to the existing source.

Since the Green’s function in homogeneous full-space is known as eikr4πr, upon applyingthe 0th-order Hankel transform, one may obtain the transformed field via

eikr

4πr

8»0

eγ|z|4πγ

J0pξρqξ dξ (B-29)

where r2 ρ2 z2 is in cylindrical coordinate. The above transform is also known asSommerfeld identity. With the aid of (B-29), the particular solutions in domain Ll arereadily obtained via

Al,ppξq 1

4πγl

, zl1 z s

Bl,ppξq 1

4πγl

, s z zl.

(B-30)

B-6

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Propagator matrix

For analytical and numerical purpose, it is useful to introduce dimensionless parameters forall the domains

ρ k0ρ, z k0z, h k0h, zj k0zj, s k0s

ξ ξk0, εj εjε0, µj µjµ0, kj kjk0

(B-31)

where ε0 and µ0 are reference permittivity and permeability, respectively; k0 ω?

ε0µ0 isreference wave number. In addition, the unknown coefficients Aj and Bj may be recast indimensionless form as

αjpξ, zq ajpξq δjlHps zqal,ppξq k0

ωµ0I∆lAj eγj zj

βjpξ, zq bjpξq δjlHpz sqbl,ppξq k0

ωµ0I∆lBj eγj zj1

(B-32)

where δjl denotes Kronecker delta. By means of (B-23), (B-31) and (B-32), the dimensionlessform of (B-24) is given by

vρ,jpξ, zq iµj

k2j

ξ g1 i

εj

ξ γj

αjpξ, zqeγjpzzjq βjpξ, zqeγjpzzj1qvz,jpξ, zq iµj

k2j

ξ2 g i

εj

ξ2αjpξ, zqeγjpzzjq βjpξ, zqeγjpzzj1q ,

(B-33)

or in matrix form via vρ,jpξ, zqvz,jpξ, zq

i

εj

ξ2 i

εj

ξ2

i

εj

ξγji

εj

ξγj

αjpξ, zqeγjpzzjq

βjpξ, zqeγjpzzj1q

(B-34)

whereas the dimensionless interfacial conditions on the jth interface is that

vρ,jpξ, zjq vρ,j1pξ, zjqεj vz,jpξ, zjq εj1vξ,z,j1pξ, zjq.

(B-35)

From (B-34), one may observe that the coefficient αj is associated with wave propagatingupward because of eγj z. Vice versa, the coefficient βj is associated with wave propagatingdownward. In addition, for the electric field due to a vertical dipole source, there are 2unknowns (α and β) in each layer. As a result, for the general n2 layered system, 2pn2q

B-7

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unknowns in total are required to be determined. With the aid of interfacial conditions in(B-35), it is possible to establish a relationship between the dimensionless coefficients in thejth layer and the adjacent layer. However, there are only n 1 interfaces which provide2pn 1q relationships, leaving the last two relationships come from the regularity conditionof the two half spaces, i.e.,

limzÑ8 vρpξ, zq 0

limzÑ8 vzpξ, zq 0.

(B-36)

Prompted by the presence of eγj zj and eγj zj in (B-33), the regularity conditions in (B-36)turns out to be that

αn1pξ, zq 0, β0pξ, zq 0. (B-37)

In the following, upon applying the interfacial conditions (B-35) to the matrix form rep-resentation (B-34) and after some manipulation, the so-called propagator matrix is obtainedvia αjpξ, zjq

βj1pξ, zjq

T uj pξq Rd

j pξqRu

j pξq T dj pξq

Ej1 0

0 Ej

αj1pξ, zjqβjpξ, zjq

(B-38)

where Ej eγj hj and denoting χj γjεj T uj pξq Rd

j pξqRu

j pξq T dj pξq

1

χj χj1

2χj1 χj χj1

χj1 χj 2χj

(B-39)

In the above, Tj and Rj are denoted as transmission and reflection coefficients of the jthinterface, respectively; the superscripts “d”(downward) and “u”(upward) denote the direc-tions of wave propagation. For example, the upward propagation wave in the jth layer is theaddition of two terms, viz, the upward propagation wave in the lower layer (j 1th) aftertransmitted through the jth interface and the downward propagation wave in the same layer(jth) after reflected from the jth interface, as shown in Fig. B-2.

With the aid of the regularity condition in (B-37) that β0pξ, zq 0, (B-38) can beexpressed in the layer j 1 as

β1pξ, z0q Ru0E1α1pξ, z0q, at z z0

α1pξ, z1q T u1 E2α2pξ, z1q Rd

1E1β1pξ, z1q, at z z1

(B-40)

Prompted by (B-32) and the fact that layer 1 contains no excitation source, one may observethat

α1pξ, z0q α1pξ, z1q α1, β1pξ, z0q β1pξ, z1q β1 (B-41)

B-8

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Figure B-2: Reflection and transmission coefficients

Therefore, (B-40) can be reformulated in a compact forms via

β1 pRu0E1α1

α1 1Rd

1pRu

0E21

1

T u1 E2α2 pT u

1 E2α2

(B-42)

Similarly, in the layer j 2, the coefficients can be formulated as

β2 Ru1E2α2 T d

1 E1β1

α2 T u2 E3α3 Rd

2E2β2

(B-43)

and the compact forms after manipulation are given via

β2 Ru

1 T d1pRu

0pT u1 E2

1

E2α2 pRu

1E2α2

α2 1Rd

2pRu

1E22

1

T u2 E3α3 pT u

2 E3α3

(B-44)

In the above, pTj and pRj are called generalized transmission and reflection coefficients whichtake account of the multiple transmissions and reflections above or under the jth layer. Infact, they can be obtained recursively viapT u

j 1Rd

jpRu

j1E2j

1

T ujpRu

j Ruj T d

jpRu

j1pT uj E2

jpRu1 0

(B-45)

B-9

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and the equation at z zl1

βlpξ, zl1q pRul1Elαlpξ, zl1q (B-46)

for all layers above the source (0 ¤ j ¤ l 1), as well aspT dj

1RujpRd

j1E2j

1

T djpRd

j Rdj T u

jpRd

j1pT dj E2

jpRdn1 0

(B-47)

and the equation at z zl

αlpξ, zlq pRdl Elβlpξ, zlq (B-48)

for all layers below the source (l 1 ¤ j ¤ n 1).For layer j l which contains the excitation source, upon substituting (B-32) into (B-46)

and (B-48) and solving for the unknowns al and bl, one may obtain the explicit expressionas

alpξq 1 pRd

lpRu

l1E2l

1 pRdl Elbl,ppξq pRd

lpRu

l1E2l al,ppξq

blpξq

1 pRdlpRu

l1E2l

1 pRul1Elal,ppξq pRu

l1pRd

l E2l bl,ppξq

(B-49)

where

al,ppξq bl,ppξq 1

4πγl

(B-50)

Therefore, the total coefficients for the source layer are given by

αlpξ, zq 1 pRd

lpRu

l1E2l

1 pRdl Elbl,p pRd

lpRu

l1E2l al,p

Hps zqal,p

βlpξ, zq 1 pRd

lpRu

l1E2l

1 pRul1Elal,p pRu

l1pRd

l E2l bl,p

Hpz sqbl,p

(B-51)

To this point, based on equations (B-38)(B-51), one may extend the derivation for thecoefficients αl and βl in the source layer to other coefficients αj and βj for the layers aboveand below the source layer.

Without losing any generality, one may assume that the source is located in the upperhalf of the source layer, i.e., zl1 s pzl1 zlq2. For z ¡ s, the lth layer coefficients

B-10

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become to that

glpξ, zq αlpξ, zqeγlpzzlq βlpξ, zqeγlpzzl1q

1

4πγl

1 pRd

lpRu

l1E2l

1

...

pRdl eγlp2zlzsq pRu

l1eγlpsz2zl1q eγlpzsq ...

pRul1

pRdl E2

l eγlpszq(B-52)

In what follows,

eγlp2zlzsq opeξlhl2q, eγlhl opeξhlq, (B-53)

when ξ Ñ 8, which results in the approximation of (B-52) by the following

glpξ, zq 1

4πγl

eγlpzsq pRu

l1eγlpzs2zl1q

. (B-54)

Furthermore, based on (B-39) and (B-45), it can be shown that

pRuj Ru

j T djpRu

j1Tuj E2

j

1RdjR

uj1E2

j

Ruj (B-55)

when ξ Ñ 8, i.e., pRuj χj1 χj

χj1 χj

(B-56)

where χj γjεj. Upon substituting (B-56) into (B-54), one may define the asymptoticterm of the total kernel as

gal pξ, zq 1

4πξ

eγlpzsq χj1 χj

χj1 χj

eγlpzs2zl1q

(B-57)

in which the superscript “a” denotes asymptotic. To simplify the formula, one may definethe relative dimensionless depths as the local variables with respect to the interface at depthzl1 as

z1 z zl1, s1 s zl1 (B-58)

and the dimensionless distance variable as

d1 |z1 s1|, d2 z1 s1. (B-59)

As a result of (B-33), the asymptotic term of the total kernel for z ¡ s in domain Ll may

B-11

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be written as

vaρ,lpξ, zq iξ

4πεl

eγld1 χl1 χl

χl1 χl

eγld2

va

z,lpξ, zq iξ2

4πεlγl

eγld1 χl1 χl

χl1 χl

eγld2

(B-60)

In the same manner, one may derive the asymptotic term for z s in domain Ll as

vaρ,lpξ, zq iξ

4πεl

eγld1 χl1 χl

χl1 χl

eγld2

va

z,lpξ, zq iξ2

4πεlγl

eγld1 χl1 χl

χl1 χl

eγld2

(B-61)

In the above, since the source is located in the upper half of the lth layer, the l 1th interface is the closest one to the source, which in turn has the largest impact on theelectromagnetic field over the other interfaces. As a result, one may consider a local systemthat includes the interface at depth zl1, the l 1th and lth layers. The majority of thewave energy is enclosed within this local system, whereas at a further distance outside thewave is exponentially vanished. In other words, the asymptotic term capture the singularityof the total kernel.

One can also show that when source is located in the lower half of the source layer, i.e.,pzl1 zlq2 s zl, the local variables are defined with respect to the interface at depthzl which is, nevertheless, the closest one to the source. In addition, instead of the l 1thlayer, the l 1th layer should be included in the local system as well as the lth layer andthe interface at depth zl.

In order to obtain the electric field in the original form, one should apply the Hankeltransform in (B-17) to the transformed fields in (B-60) and (B-61). However, due to therapid oscillation of the Bessel function when its variable is at 8, the success of a fastintegration may be limited for a conventional numerical integration scheme.

To expedite the integration, first of all, the Eq. (B-60) and (B-61) in which the integrandmay be considered as a function of ξ only shall be rephrased here

vaρ,1pξq iξ

4πε1

eγ1d1 χ2 χ1

χ2 χ1

eγ1d2

va

z,1pξq iξ2

4πε1γ1

eγ1d1 χ2 χ1

χ2 χ1

eγ1d2

.

(B-62)

where the domain contains the source is labeled as “1” and the other adjacent layer as “2”;all the layer properties are labeled with respect to the layer they belong to.

The first terms inside the parenthesis may be considered separately, whose integrations,

B-12

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denoted as J γ1 pzq and J γ

0 pzq, can be obtained in closed-form via

J γ1 pzq

8»0

eγz ξ2 J1pξρq dξ zρk2

r3 3izρk

r4 3zρ

r5

eikr

J γ0 pzq

8»0

eγz ξ3

γJ0pξρq dξ

k2ρ2

r3 ikp2z2 ρ2q

r4 z2 ρ2

r5

eikr

(B-63)

where r2 z2 ρ2 and k2 γ2 ξ2, which leaves the computation to the evaluation of thesecond terms inside the parenthesis.

To the second terms, the closed-form integrations are not available, but available to theasymptote of the integrand as will be illustrated below. In what follows, due to the fact thatBessel function is bounded around zero,

limxÑ0

J0pxq Ñ 1, limxÑ0

J1pxq Ñ 0 (B-64)

the integrands are bounded around zero as well. When ξ Ñ 8, the integrands of the secondterms can be expanded as the following

ε2γ1 ε1γ2

ε2γ1 ε1γ2

ξ2eγ1z cρ,2pzq ξ2 cρ,1pzq ξ cρ,0pzq opξ1q eξz

ε2γ1 ε1γ2

ε2γ1 ε1γ2

ξ3

γ2

eγ1z cz,2pzq ξ2 cz,1pzq ξ cz,0pzq opξ1q eξz

(B-65)

where

cρ,2pzq cz,2pzq ε1 ε2

ε1 ε2

cρ,1pzq cz,1pzq ε1pε1 ε2q2pε1 ε2q z

cρ,0pzq ε21pε2

1 ε22qz2 8ε1ε2pε1 ε2q8pε1 ε2q2

cz,0pzq ε21pε2

1 ε22qz2 4ε1pε1 ε2qpε1 3ε2q

8pε1 ε2q2

(B-66)

It is useful to denote

Jvpn, zq 8»0

ξneξzJvpξρqdξ (B-67)

where v 0, 1 and n 0, 1, 2. The closed-form of the above is available in the table of

B-13

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integral transform as shown below

J1p2, zq 3zρ

r5

J1p1, zq ρ

r3

J1p0, zq ρ

rpz rq

J0p2, zq 2z2 ρ2

r5

J0p1, zq z

r3

J0p0, zq 1

r.

(B-68)

To ensure the availability of the closed-form for the integration of the asymptotic terms,the previous derivations in (B-62) may be extended to a further step as

vaρ,1pξq i

4πε1

ξeγ1d1 cρ,2pzqξ cρ,1pzq cρ,0pzqξ1

eξd2

va

z,1pξq i

4πε1

ξ2

γ1

eγ1d1 cz,2pzqξ cz,1pzq cz,0pzqξ1

eξd2

(B-69)

and the residual terms are given as

vrρ,1pξq vρ,1pξq va

ρ,1pξqvr

z,1pξq vz,1pξq vaz,1pξq.

(B-70)

Due to the fast decay of the residual terms when ξ Ñ 8, it allows the fast evaluation of theintegral of the residual terms by truncating along the integration path.

With the aid of (B-63), (B-67) and (B-68), the closed-form for the asymptotic terms canbe obtained via

vaρ,1pρ, zq

8»0

vaρ,1pξqξJ1pξρqdξ

i

4πε1

J γ

1 pzq 2

m0

cρ,mpzqJ1pm, zq

vaz,1pρ, zq

8»0

vaz,1pξqξJ0pξρqdξ

i

4πε1

J γ

0 pzq 2

m0

cz,mpzqJ0pm, zq

(B-71)

B-14

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To this point, the dimensional electric fields in the source layer are obtained via

Eρ,1pρ, zq ωµ0I∆l

k0

8»0

va

ρ,1 vrρ,1

ξJ1pρqdξ ωµ0I∆l

k0

pvρ,1qa pvρ,1qr

Ez,1pρ, zq ωµ0I∆l

k0

8»0

va

z,1 vrz,1

ξJ0pρqdξ ωµ0I∆l

k0

pvz,1qa pvz,1qr

(B-72)

where the closed-form are available to the asymptotic terms; the residual terms are evaluatedin a truncated integration. Upon solving the coefficients in the layers other than the sourcelayer through the recursive propagator matrixes, the dimensional electric fields in the otherlayer can be readily obtained.

B-15

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Appendix CManual for GPR interpretation

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To facilitate the implementation of this method, a graphic user interface is designed forconvenient operation, called “GopherGPR” as shown in Fig. C-1.

Step 1: Input the scan number where the program will start to search for amatched scan and click the button “Step 1” to locate the “DZT” filesfor the metal calibration and the pavement survey.

Upon clicking the button “Step 1”, the user is asked to browse for and locate the “.DZT”RADAN files for the metal calibration first, and then for the pavement survey. In this case,the metal calibration file is named “Metal1001.DZT”. After selecting the files, the programwill search for scans starts from the chosen scan number within the pavement survey ”DZT”file. The program will stop searching after it finds the first scan which matches the metalplate calibration in terms of the effective height. The option to search all the scans willbe available later. If there are no matched scans in the whole file, the program will beterminated, showing “Height Mismatch” in the text field below.

The value of “effective H” is shown as 25.9 inches, which in return means the nominalheight of the GPR box is around 19 inches above the ground. The value of this effectiveheight is associated with the metal calibration file “Metal1001.DZT”, which is calculatedaccording to the “Handbook for GPR Inspection of Road Structures” from GeophysicalSurvey Systems, Inc (GSSI). The value of “applied H” in the popup window is the one usedin the model which corresponds to the effective height of the metal calibration scan. Note thatthe value of the “applied H” may be only applicable to the sample metal calibration providedin this project. For a new metal calibration whose effective height may be different than25.9 inches, the model will need to be adjusted prior to the interpretation. Furthermore, itshould be pointed out that each different calibration height is referred to a differently trainedneural networks.

If a matched scan is found, the scans of the metal plate calibration and pavement surveyare plotting in the top two windows on the panel, respectively. The program will also plotthe metal plate calibration scan (in blue) and the pavement survey scan (in green) togetherfrom the effective time-zero in the window called “Pavement & Metal”. The subtraction ofthe metal calibration from the pavement survey is also plotted in the window (in blue) called“Diff=Pavement-Metal”, where the user can check whether the unwanted pulse from the feedpoint is removed. The red line is an illustration from the simulation data, which is also thesubtraction of the metal plate simulation from a simulation of a typical pavement profile. Aclose “fit” or overlap of the red and blue lines indicates that the height of effective time-zeroof the survey matches the height of the simulation. Just to the right of this window, asquared window shows the matched reflection pulse of the survey (in blue) and simulation(in red) (similar to Fig. 6.2).

Step 2: Adjust the source pattern of the assumed Mexican-hat pulse (optional).

Two slider bars are designed, One for the time shift, and the other one for the pulseparameter R which controls the width of the pulse. The user may move the slider bars to

C-1

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Figure C-1: GopherGPR panel

C-2

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adjust the pattern of the source. In the end, the reflection of the simulated pulse (in red)should closely overlap with that of the survey pulse (in blue).

In addition, the program provides a cleanup scheme to reduce the extra noise at thebeginning and ending of the scan. Under the panel “Sample #”, the first sample numberis the location before which the amplitudes are set to zero. Moreover, in order to suppressthe noise at the end of the scan in certain occasions, the amplitudes after the location of thesecond sample number will be set to zero as well. Since the capability of the interpretationis limited to the system with asphalt less than 10 inches, the position of the second samplenumber may be also used to suppress the reflections from additional or deeper interfaces,and the ambient noise as well.

Step 3: Click the button “Step 2” which plots the FRF of the subtraction ac-cording to Eq. 6.3 in the window called “FRF pattern”.

The upper one is the real portion, and the lower one is the imaginary portion. The rangeof the FRF (0.95GHz1.05GHz) and the number of FRF segments will be displayed in thetext fields after clicking the button “Step 2”.

Step 4: Click the button “Step 3” to interpret the pavement profile from thecomputed FRF pattern.

The outputs are shown in the text fields above the button “Step 3”. From top to bottom,they are the dielectric constant of the asphalt layer, the thickness of the asphalt layer andthe dielectric constant of base layer. If the “Profile” box is checked, the road profile will beobtained between the scan numbers input by user.

C-3